gdz.top
Номер 378, страница 116 - гдз по геометрии 11 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2020 - 2026
ISBN: 978-601-317-528-7
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 11 классе
III. Тела вращения и их элементы. 17. Площадь поверхности усеченного конуса - номер 378, страница 116.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144589 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Тела вращения и их элементы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Book {#1056 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:1 [ 0 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1063 #items: array:1 [ 0 => App\Models\Term {#1060 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1070 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:3 [ 0 => App\Models\Term {#1076 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "field_surname_rp" => null ] #original: array:12 [ "id" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1077 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1064 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Term {#1072 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Term {#1068 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 543 "class_subject" => 393 ] ] #original: array:50 [ "id" => 4313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1061} "field_class" => Illuminate\Database\Eloquent\Collection {#1063} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071} "field_author" => Illuminate\Database\Eloquent\Collection {#1062} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1073} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1074} "field_country" => Illuminate\Database\Eloquent\Collection {#1067} "field_city" => Illuminate\Database\Eloquent\Collection {#1065} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1079} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1080} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1081} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1082} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1083} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1084} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1086} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 543 "class_subject" => 393 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Тела вращения и их элементы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1057} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1049} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:2 [ 0 => App\Models\Branch {#1116 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Тела вращения и их элементы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "III. Тела вращения и их элементы" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1117} "field_page_start" => "88" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1118} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1120 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144184 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "17. Площадь поверхности усеченного конуса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "113" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144184 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "17. Площадь поверхности усеченного конуса" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "113" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1122} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1123} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:3 [ 0 => App\Models\Element {#1098 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1379911 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1070} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1041 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1051} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1041} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1052} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "text" => "<p><strong>378.</strong> Какую высоту будет иметь ведро, если в заготовке для получения его боковой поверхности величины дуг равны по $60^\circ$, а их радиусы – $72 \text{ см}$ и $48 \text{ см}$? Ответ дайте с точностью до $0,1 \text{ см}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "378-1.jpg" "alt" => null "width" => "1551" "height" => 267 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/378-1.webp?ts=1754600466" ] ] ] #original: array:7 [ "id" => 1379911 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1088} "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "text" => "<p><strong>378.</strong> Какую высоту будет иметь ведро, если в заготовке для получения его боковой поверхности величины дуг равны по $60^\circ$, а их радиусы – $72 \text{ см}$ и $48 \text{ см}$? Ответ дайте с точностью до $0,1 \text{ см}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "378-1.jpg" "alt" => null "width" => "1551" "height" => 267 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/378-1.webp?ts=1754600466" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1096 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1380448 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Edition {#1095 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Term {#1097 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1092} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1094} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "378-1.jpg" "alt" => null "width" => "1046" "height" => 392 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/378-1.webp?ts=1754600525" ] ] ] #original: array:6 [ "id" => 1380448 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1050} "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "378-1.jpg" "alt" => null "width" => "1046" "height" => 392 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/378-1.webp?ts=1754600525" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1087 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1562683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 #items: array:1 [ 0 => App\Models\Edition {#1048 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1107 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1043} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1107} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Угол дуг $\alpha = 60^\circ$</p><p>Радиусы заготовки $L_1 = 72 \text{ см}$, $L_2 = 48 \text{ см}$</p><p><strong>Перевод в СИ:</strong></p><p>$\alpha = 60^\circ = 60 \cdot \frac{\pi}{180} \text{ рад} = \frac{\pi}{3} \text{ рад}$</p><p>$L_1 = 72 \text{ см} = 0.72 \text{ м}$</p><p>$L_2 = 48 \text{ см} = 0.48 \text{ м}$</p><p><strong>Найти:</strong></p><p>Высота ведра $h$</p><p><strong>Решение:</strong></p><p>Боковая поверхность ведра представляет собой усеченный конус. Заготовка для его боковой поверхности является частью кольца (сектором кольца). Радиусы дуг заготовки ($L_1$ и $L_2$) являются образующими полного и усеченного конусов, соответственно. Угол дуг ($\alpha$) определяет длины окружностей оснований ведра.</p><p><strong>1.</strong> <strong>Определение радиусов оснований ведра ($r_1, r_2$):</strong></p><p>Длина большей дуги заготовки становится длиной окружности большего основания ведра. Длина меньшей дуги заготовки становится длиной окружности меньшего основания ведра. Формула длины дуги: $S = \frac{\alpha}{360^\circ} \cdot 2\pi R$.</p><p>Для большего основания (радиус $r_1$):<br>Длина дуги $C_1 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_1 = \frac{1}{6} \cdot 2\pi (72 \text{ см}) = 24\pi \text{ см}$.<br>Эта длина равна $2\pi r_1$, где $r_1$ — радиус большего основания ведра.<br>$2\pi r_1 = 24\pi \text{ см} \implies r_1 = 12 \text{ см}$.</p><p>Для меньшего основания (радиус $r_2$):<br>Длина дуги $C_2 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_2 = \frac{1}{6} \cdot 2\pi (48 \text{ см}) = 16\pi \text{ см}$.<br>Эта длина равна $2\pi r_2$, где $r_2$ — радиус меньшего основания ведра.<br>$2\pi r_2 = 16\pi \text{ см} \implies r_2 = 8 \text{ см}$.</p><p><strong>2.</strong> <strong>Определение образующей ведра ($l$):</strong></p><p>Образующая ведра является разницей между радиусами дуг заготовки.<br>$l = L_1 - L_2 = 72 \text{ см} - 48 \text{ см} = 24 \text{ см}$.</p><p><strong>3.</strong> <strong>Определение высоты ведра ($h$):</strong></p><p>Высоту ведра можно найти по теореме Пифагора, рассматривая прямоугольный треугольник, образованный высотой ведра, разностью радиусов оснований и образующей.<br>$h^2 + (r_1 - r_2)^2 = l^2$.<br>Отсюда $h = \sqrt{l^2 - (r_1 - r_2)^2}$.<br>Разность радиусов: $r_1 - r_2 = 12 \text{ см} - 8 \text{ см} = 4 \text{ см}$.<br>Подставляем значения:<br>$h = \sqrt{(24 \text{ см})^2 - (4 \text{ см})^2}$.<br>$h = \sqrt{576 \text{ см}^2 - 16 \text{ см}^2}$.<br>$h = \sqrt{560 \text{ см}^2}$.<br>$h \approx 23.664319 \text{ см}$.</p><p><strong>4.</strong> <strong>Округление результата:</strong></p><p>Требуется округлить ответ до 0.1 см.<br>$h \approx 23.7 \text{ см}$.</p><p><strong>Ответ:</strong> $23.7 \text{ см}$.</p>" ] #original: array:6 [ "id" => 1562683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109} "task" => array:2 [ "refs" => "1144589" "type" => "task" ] "text" => "<p><strong>Дано:</strong></p><p>Угол дуг $\alpha = 60^\circ$</p><p>Радиусы заготовки $L_1 = 72 \text{ см}$, $L_2 = 48 \text{ см}$</p><p><strong>Перевод в СИ:</strong></p><p>$\alpha = 60^\circ = 60 \cdot \frac{\pi}{180} \text{ рад} = \frac{\pi}{3} \text{ рад}$</p><p>$L_1 = 72 \text{ см} = 0.72 \text{ м}$</p><p>$L_2 = 48 \text{ см} = 0.48 \text{ м}$</p><p><strong>Найти:</strong></p><p>Высота ведра $h$</p><p><strong>Решение:</strong></p><p>Боковая поверхность ведра представляет собой усеченный конус. Заготовка для его боковой поверхности является частью кольца (сектором кольца). Радиусы дуг заготовки ($L_1$ и $L_2$) являются образующими полного и усеченного конусов, соответственно. Угол дуг ($\alpha$) определяет длины окружностей оснований ведра.</p><p><strong>1.</strong> <strong>Определение радиусов оснований ведра ($r_1, r_2$):</strong></p><p>Длина большей дуги заготовки становится длиной окружности большего основания ведра. Длина меньшей дуги заготовки становится длиной окружности меньшего основания ведра. Формула длины дуги: $S = \frac{\alpha}{360^\circ} \cdot 2\pi R$.</p><p>Для большего основания (радиус $r_1$):<br>Длина дуги $C_1 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_1 = \frac{1}{6} \cdot 2\pi (72 \text{ см}) = 24\pi \text{ см}$.<br>Эта длина равна $2\pi r_1$, где $r_1$ — радиус большего основания ведра.<br>$2\pi r_1 = 24\pi \text{ см} \implies r_1 = 12 \text{ см}$.</p><p>Для меньшего основания (радиус $r_2$):<br>Длина дуги $C_2 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_2 = \frac{1}{6} \cdot 2\pi (48 \text{ см}) = 16\pi \text{ см}$.<br>Эта длина равна $2\pi r_2$, где $r_2$ — радиус меньшего основания ведра.<br>$2\pi r_2 = 16\pi \text{ см} \implies r_2 = 8 \text{ см}$.</p><p><strong>2.</strong> <strong>Определение образующей ведра ($l$):</strong></p><p>Образующая ведра является разницей между радиусами дуг заготовки.<br>$l = L_1 - L_2 = 72 \text{ см} - 48 \text{ см} = 24 \text{ см}$.</p><p><strong>3.</strong> <strong>Определение высоты ведра ($h$):</strong></p><p>Высоту ведра можно найти по теореме Пифагора, рассматривая прямоугольный треугольник, образованный высотой ведра, разностью радиусов оснований и образующей.<br>$h^2 + (r_1 - r_2)^2 = l^2$.<br>Отсюда $h = \sqrt{l^2 - (r_1 - r_2)^2}$.<br>Разность радиусов: $r_1 - r_2 = 12 \text{ см} - 8 \text{ см} = 4 \text{ см}$.<br>Подставляем значения:<br>$h = \sqrt{(24 \text{ см})^2 - (4 \text{ см})^2}$.<br>$h = \sqrt{576 \text{ см}^2 - 16 \text{ см}^2}$.<br>$h = \sqrt{560 \text{ см}^2}$.<br>$h \approx 23.664319 \text{ см}$.</p><p><strong>4.</strong> <strong>Округление результата:</strong></p><p>Требуется округлить ответ до 0.1 см.<br>$h \approx 23.7 \text{ см}$.</p><p><strong>Ответ:</strong> $23.7 \text{ см}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144590" "type" => "task" ] "previous" => array:2 [ "refs" => "1144588" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\BookPage {#1039 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1116746 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/page-116" "field_display_title" => "116" "field_folder" => "1" "field_image_name" => "116" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Element {#1130 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1295011 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Edition {#1090} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "116-1.jpg" "alt" => null "width" => "2008" "height" => 2972 "path" => "/media/geometrija_11/Kokshetau-u20/0-1/116-1.webp?ts=1753338810" ] ] ] #original: array:6 [ "id" => 1295011 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132} "book_page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "116-1.jpg" "alt" => null "width" => "2008" "height" => 2972 "path" => "/media/geometrija_11/Kokshetau-u20/0-1/116-1.webp?ts=1753338810" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1116747" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116745" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:5 [ 0 => App\Models\Task {#1210 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144589 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Branch {#1116} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:2 [ 0 => App\Models\Branch {#1116} 1 => App\Models\Branch {#1120} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:3 [ 0 => App\Models\Element {#1098} 1 => App\Models\Element {#1096} 2 => App\Models\Element {#1087} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144590" "type" => "task" ] "previous" => array:2 [ "refs" => "1144588" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144589 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1211} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1212} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1213} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1214} "content" => Illuminate\Database\Eloquent\Collection {#1215} "next" => array:2 [ "refs" => "1144590" "type" => "task" ] "previous" => array:2 [ "refs" => "1144588" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1216} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1217 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144590 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-379" "field_display_title" => "379" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:1 [ 0 => App\Models\Branch {#1116} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:2 [ 0 => App\Models\Branch {#1116} 1 => App\Models\Branch {#1120} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:3 [ 0 => App\Models\Element {#1233 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1235 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1237 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144591" "type" => "task" ] "previous" => array:2 [ "refs" => "1144589" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144590 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-379" "field_display_title" => "379" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1219} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1220} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221} "content" => Illuminate\Database\Eloquent\Collection {#1224} "next" => array:2 [ "refs" => "1144591" "type" => "task" ] "previous" => array:2 [ "refs" => "1144589" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1230} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1223 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-380" "field_display_title" => "380" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1229 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:1 [ 0 => App\Models\Branch {#1116} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:2 [ 0 => App\Models\Branch {#1116} 1 => App\Models\Branch {#1120} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:3 [ 0 => App\Models\Element {#1247 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Element {#1249 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Element {#1251 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144592" "type" => "task" ] "previous" => array:2 [ "refs" => "1144590" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1244 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144591 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-380" "field_display_title" => "380" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1229} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1227} "content" => Illuminate\Database\Eloquent\Collection {#1226} "next" => array:2 [ "refs" => "1144592" "type" => "task" ] "previous" => array:2 [ "refs" => "1144590" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1244} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1228 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-381" "field_display_title" => "381" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1243 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:1 [ 0 => App\Models\Branch {#1116} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:2 [ 0 => App\Models\Branch {#1116} 1 => App\Models\Branch {#1120} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1240 #items: array:3 [ 0 => App\Models\Element {#1261 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] …10 } 1 => App\Models\Element {#1263 …30} 2 => App\Models\Element {#1265 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144593" "type" => "task" ] "previous" => array:2 [ "refs" => "1144591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144592 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-381" "field_display_title" => "381" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1243} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1241} "content" => Illuminate\Database\Eloquent\Collection {#1240} "next" => array:2 [ "refs" => "1144593" "type" => "task" ] "previous" => array:2 [ "refs" => "1144591" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1258} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1242 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-382" "field_display_title" => "382" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1257 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1259 #items: array:1 [ 0 => App\Models\Branch {#1116} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:2 [ 0 => App\Models\Branch {#1116} 1 => App\Models\Branch {#1120} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:3 [ 0 => App\Models\Element {#1275 …30} 1 => App\Models\Element {#1277 …30} 2 => App\Models\Element {#1279 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144594" "type" => "task" ] "previous" => array:2 [ "refs" => "1144592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ 0 => App\Models\Book {#1056} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144593 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-382" "field_display_title" => "382" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1239} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1257} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1259} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1255} "content" => Illuminate\Database\Eloquent\Collection {#1254} "next" => array:2 [ "refs" => "1144594" "type" => "task" ] "previous" => array:2 [ "refs" => "1144592" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1272} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1116746 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/page-116" "field_display_title" => "116" "field_folder" => "1" "field_image_name" => "116" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1115} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1111} "content" => Illuminate\Database\Eloquent\Collection {#1131} "next" => array:2 [ "refs" => "1116747" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116745" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144589 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "116" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-378" "field_display_title" => "378" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1108} "content" => Illuminate\Database\Eloquent\Collection {#1105} "next" => array:2 [ "refs" => "1144590" "type" => "task" ] "previous" => array:2 [ "refs" => "1144588" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1101} "page" => array:2 [ "refs" => "1116746" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№378 (с. 116)
Условие. №378 (с. 116)
скриншот условия
378. Какую высоту будет иметь ведро, если в заготовке для получения его боковой поверхности величины дуг равны по $60^\circ$, а их радиусы – $72 \text{ см}$ и $48 \text{ см}$? Ответ дайте с точностью до $0,1 \text{ см}$.
Решение. №378 (с. 116)
Решение 2 (rus). №378 (с. 116)
Дано:
Угол дуг $\alpha = 60^\circ$
Радиусы заготовки $L_1 = 72 \text{ см}$, $L_2 = 48 \text{ см}$
Перевод в СИ:
$\alpha = 60^\circ = 60 \cdot \frac{\pi}{180} \text{ рад} = \frac{\pi}{3} \text{ рад}$
$L_1 = 72 \text{ см} = 0.72 \text{ м}$
$L_2 = 48 \text{ см} = 0.48 \text{ м}$
Найти:
Высота ведра $h$
Решение:
Боковая поверхность ведра представляет собой усеченный конус. Заготовка для его боковой поверхности является частью кольца (сектором кольца). Радиусы дуг заготовки ($L_1$ и $L_2$) являются образующими полного и усеченного конусов, соответственно. Угол дуг ($\alpha$) определяет длины окружностей оснований ведра.
1. Определение радиусов оснований ведра ($r_1, r_2$):
Длина большей дуги заготовки становится длиной окружности большего основания ведра. Длина меньшей дуги заготовки становится длиной окружности меньшего основания ведра. Формула длины дуги: $S = \frac{\alpha}{360^\circ} \cdot 2\pi R$.
Для большего основания (радиус $r_1$):
Длина дуги $C_1 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_1 = \frac{1}{6} \cdot 2\pi (72 \text{ см}) = 24\pi \text{ см}$.
Эта длина равна $2\pi r_1$, где $r_1$ — радиус большего основания ведра.
$2\pi r_1 = 24\pi \text{ см} \implies r_1 = 12 \text{ см}$.
Для меньшего основания (радиус $r_2$):
Длина дуги $C_2 = \frac{60^\circ}{360^\circ} \cdot 2\pi L_2 = \frac{1}{6} \cdot 2\pi (48 \text{ см}) = 16\pi \text{ см}$.
Эта длина равна $2\pi r_2$, где $r_2$ — радиус меньшего основания ведра.
$2\pi r_2 = 16\pi \text{ см} \implies r_2 = 8 \text{ см}$.
2. Определение образующей ведра ($l$):
Образующая ведра является разницей между радиусами дуг заготовки.
$l = L_1 - L_2 = 72 \text{ см} - 48 \text{ см} = 24 \text{ см}$.
3. Определение высоты ведра ($h$):
Высоту ведра можно найти по теореме Пифагора, рассматривая прямоугольный треугольник, образованный высотой ведра, разностью радиусов оснований и образующей.
$h^2 + (r_1 - r_2)^2 = l^2$.
Отсюда $h = \sqrt{l^2 - (r_1 - r_2)^2}$.
Разность радиусов: $r_1 - r_2 = 12 \text{ см} - 8 \text{ см} = 4 \text{ см}$.
Подставляем значения:
$h = \sqrt{(24 \text{ см})^2 - (4 \text{ см})^2}$.
$h = \sqrt{576 \text{ см}^2 - 16 \text{ см}^2}$.
$h = \sqrt{560 \text{ см}^2}$.
$h \approx 23.664319 \text{ см}$.
4. Округление результата:
Требуется округлить ответ до 0.1 см.
$h \approx 23.7 \text{ см}$.
Ответ: $23.7 \text{ см}$.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 11 класс, для упражнения номер 378 расположенного на странице 116 к учебнику 2020 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по геометрии к упражнению №378 (с. 116), авторов: Солтан (Г Н), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.