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Вопросы?, страница 97 - 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. 13. Площадь поверхности цилиндра - страница 97.
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" => 1144514 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "97" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/vopr-13" "field_display_title" => "Вопросы?" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #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 {#1034 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Branch {#1044 #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 {#1046 #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 {#1056 #items: array:1 [ 0 => App\Models\Book {#1055 #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 {#1060 #items: array:1 [ 0 => App\Models\Term {#1057 #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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1059 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1061 #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 {#1067 #items: array:3 [ 0 => App\Models\Term {#1063 #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 {#1065 #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 {#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: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 {#1068 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Term {#1066 #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 {#1073 #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" => 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 {#1075 #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" => 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 {#1074 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1077 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1079 #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 {#1080 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1083 #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 {#1060} "field_class" => Illuminate\Database\Eloquent\Collection {#1062} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1064} "field_author" => Illuminate\Database\Eloquent\Collection {#1067} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1068} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_country" => Illuminate\Database\Eloquent\Collection {#1073} "field_city" => Illuminate\Database\Eloquent\Collection {#1075} "field_series" => Illuminate\Database\Eloquent\Collection {#1074} "field_umk" => Illuminate\Database\Eloquent\Collection {#1069} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1076} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1077} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1078} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1079} "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 {#1080} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1083} "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 {#1048 #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 {#1046} "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 {#1056} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #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 {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #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 {#1114 #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 {#1115 #items: array:1 [ 0 => App\Models\Book {#1055} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #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 {#1114} "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 {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #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 {#1117 #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" => 1144180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "13. Площадь поверхности цилиндра" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "95" "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 {#1119 #items: array:1 [ 0 => App\Models\Book {#1055} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Branch {#1044} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144180 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "13. Площадь поверхности цилиндра" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "95" "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 {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #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 {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #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" => 1380480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1087 #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1061} ] #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 {#1040 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053 #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 {#1050} "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 {#1040} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1051} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1053} "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" => "1144514" "type" => "task" ] "text" => "<p><strong>ВОПРОСЫ</strong></p><p><strong>1.</strong> Что принимается за площадь поверхности цилиндра?</p><p><strong>2.</strong> По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.</p><p><strong>3.</strong> Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1566" "height" => 490 "path" => "/media/geometrija_11/Kokshetau-u20/0-vopr/13-1.webp?ts=1754600653" ] ] ] #original: array:7 [ "id" => 1380480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1144514" "type" => "task" ] "text" => "<p><strong>ВОПРОСЫ</strong></p><p><strong>1.</strong> Что принимается за площадь поверхности цилиндра?</p><p><strong>2.</strong> По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.</p><p><strong>3.</strong> Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1566" "height" => 490 "path" => "/media/geometrija_11/Kokshetau-u20/0-vopr/13-1.webp?ts=1754600653" ] ] ] #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 {#1093 #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" => 1380764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Edition {#1049 #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 {#1103 #items: array:1 [ 0 => App\Models\Term {#1039 #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" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "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" => "gdz" ] #original: array:6 [ "id" => 6714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gdz" "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" => "gdz" ] #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_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 {#1109 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112 #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 {#1103} "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 {#1109} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1042} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1112} "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" => "1144514" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1812" "height" => 1591 "path" => "/media/geometrija_11/Kokshetau-u20/1-vopr/13-1.webp?ts=1754600709" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1659" "height" => 1115 "path" => "/media/geometrija_11/Kokshetau-u20/1-vopr/13-2.webp?ts=1754600709" ] ] ] #original: array:6 [ "id" => 1380764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1052} "task" => array:2 [ "refs" => "1144514" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "13-1.jpg" "alt" => null "width" => "1812" "height" => 1591 "path" => "/media/geometrija_11/Kokshetau-u20/1-vopr/13-1.webp?ts=1754600709" ] 1 => array:5 [ "name" => "13-2.jpg" "alt" => null "width" => "1659" "height" => 1115 "path" => "/media/geometrija_11/Kokshetau-u20/1-vopr/13-2.webp?ts=1754600709" ] ] ] #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 {#1094 #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" => 1562970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Edition {#1107 #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 {#1125 #items: array:1 [ 0 => App\Models\Term {#1121 #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" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "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" => "gemini 2.5 pro" ] #original: array:6 [ "id" => 7012 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Gemini 2.5 Pro" "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" => "gemini 2.5 pro" ] #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_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 {#1129 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126 #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 {#1125} "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 {#1129} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1126} "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" => "1144514" "type" => "task" ] "text" => "<p><strong>1.</strong> Что принимается за площадь поверхности цилиндра?</p><p>За площадь полной поверхности цилиндра принимается сумма площадей его боковой поверхности и двух его оснований. Если в задаче говорится просто о «площади поверхности цилиндра» без уточнений, как правило, имеется в виду именно площадь полной поверхности.<br><br>Ответ: Сумма площадей боковой поверхности и двух оснований цилиндра.</p><p><strong>2.</strong> По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.</p><p>Пусть дан цилиндр с радиусом основания $r$ и высотой $h$.<br><br><strong>Вывод формулы площади боковой поверхности ($S_{бок}$)</strong><br>Разверткой боковой поверхности цилиндра является прямоугольник. Одна сторона этого прямоугольника равна высоте цилиндра $h$. Другая сторона равна длине окружности основания цилиндра, которая вычисляется по формуле $C = 2\pi r$.<br>Площадь этого прямоугольника, а следовательно, и площадь боковой поверхности цилиндра, равна произведению его сторон:<br>$S_{бок} = C \cdot h = 2\pi r \cdot h = 2\pi rh$<br><br><strong>Вывод формулы площади полной поверхности ($S_{полн}$)</strong><br>Площадь полной поверхности цилиндра ($S_{полн}$) равна сумме площади его боковой поверхности ($S_{бок}$) и площадей двух его оснований ($S_{осн}$).<br>Основанием цилиндра является круг радиусом $r$. Площадь одного такого круга равна:<br>$S_{осн} = \pi r^2$<br>Поскольку у цилиндра два основания, их общая площадь составляет $2 \cdot S_{осн} = 2\pi r^2$.<br>Сложив площадь боковой поверхности и площади двух оснований, получаем формулу для площади полной поверхности:<br>$S_{полн} = S_{бок} + 2S_{осн} = 2\pi rh + 2\pi r^2$<br>Эту формулу также можно записать, вынеся за скобки общий множитель $2\pi r$:<br>$S_{полн} = 2\pi r(h + r)$<br><br>Ответ: Площадь боковой поверхности: $S_{бок} = 2\pi rh$. Площадь полной поверхности: $S_{полн} = 2\pi rh + 2\pi r^2 = 2\pi r(h + r)$.</p><p><strong>3.</strong> Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.</p><p>Разверткой боковой поверхности цилиндра является прямоугольник, одна сторона которого равна высоте цилиндра ($h$), а другая — длине окружности его основания ($2\pi r$).<br>Развертка полной поверхности цилиндра состоит из этого прямоугольника и двух кругов, которые являются основаниями цилиндра.<br><br><strong>Изображение развертки полной поверхности цилиндра:</strong><br><svg width="320" height="220" xmlns="http://www.w3.org/2000/svg" font-family="sans-serif" font-size="16px"><style> .label { font-style: italic; } </style><!-- Прямоугольник (боковая поверхность) --><rect x="60" y="60" width="200" height="100" fill="#f0f0f0" stroke="#333" stroke-width="1.5"/><!-- Верхний круг (основание) --><circle cx="160" cy="60" r="31.83" fill="#d0e0ff" stroke="#333" stroke-width="1.5"/><!-- Нижний круг (основание) --><circle cx="160" cy="160" r="31.83" fill="#d0e0ff" stroke="#333" stroke-width="1.5"/><!-- Обозначения --><!-- Высота h --><line x1="270" y1="60" x2="270" y2="160" stroke="#333" stroke-width="1"/><line x1="267" y1="60" x2="273" y2="60" stroke="#333" stroke-width="1"/><line x1="267" y1="160" x2="273" y2="160" stroke="#333" stroke-width="1"/><text x="278" y="115" class="label">h</text><!-- Длина 2πr --><line x1="60" y1="40" x2="260" y2="40" stroke="#333" stroke-width="1"/><line x1="60" y1="37" x2="60" y2="43" stroke="#333" stroke-width="1"/><line x1="260" y1="37" x2="260" y2="43" stroke="#333" stroke-width="1"/><text x="145" y="32" class="label">2πr</text><!-- Радиус r --><line x1="160" y1="160" x2="191.83" y2="160" stroke="#333" stroke-width="1" stroke-dasharray="3,2"/><text x="170" y="180" class="label">r</text></svg><br><br>Ответ: Разверткой боковой поверхности цилиндра является прямоугольник. Развертка полной поверхности, состоящая из прямоугольника (боковой поверхности) и двух кругов (оснований), изображена выше.</p>" ] #original: array:6 [ "id" => 1562970 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1106} "task" => array:2 [ "refs" => "1144514" "type" => "task" ] "text" => "<p><strong>1.</strong> Что принимается за площадь поверхности цилиндра?</p><p>За площадь полной поверхности цилиндра принимается сумма площадей его боковой поверхности и двух его оснований. Если в задаче говорится просто о «площади поверхности цилиндра» без уточнений, как правило, имеется в виду именно площадь полной поверхности.<br><br>Ответ: Сумма площадей боковой поверхности и двух оснований цилиндра.</p><p><strong>2.</strong> По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.</p><p>Пусть дан цилиндр с радиусом основания $r$ и высотой $h$.<br><br><strong>Вывод формулы площади боковой поверхности ($S_{бок}$)</strong><br>Разверткой боковой поверхности цилиндра является прямоугольник. Одна сторона этого прямоугольника равна высоте цилиндра $h$. Другая сторона равна длине окружности основания цилиндра, которая вычисляется по формуле $C = 2\pi r$.<br>Площадь этого прямоугольника, а следовательно, и площадь боковой поверхности цилиндра, равна произведению его сторон:<br>$S_{бок} = C \cdot h = 2\pi r \cdot h = 2\pi rh$<br><br><strong>Вывод формулы площади полной поверхности ($S_{полн}$)</strong><br>Площадь полной поверхности цилиндра ($S_{полн}$) равна сумме площади его боковой поверхности ($S_{бок}$) и площадей двух его оснований ($S_{осн}$).<br>Основанием цилиндра является круг радиусом $r$. Площадь одного такого круга равна:<br>$S_{осн} = \pi r^2$<br>Поскольку у цилиндра два основания, их общая площадь составляет $2 \cdot S_{осн} = 2\pi r^2$.<br>Сложив площадь боковой поверхности и площади двух оснований, получаем формулу для площади полной поверхности:<br>$S_{полн} = S_{бок} + 2S_{осн} = 2\pi rh + 2\pi r^2$<br>Эту формулу также можно записать, вынеся за скобки общий множитель $2\pi r$:<br>$S_{полн} = 2\pi r(h + r)$<br><br>Ответ: Площадь боковой поверхности: $S_{бок} = 2\pi rh$. Площадь полной поверхности: $S_{полн} = 2\pi rh + 2\pi r^2 = 2\pi r(h + r)$.</p><p><strong>3.</strong> Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.</p><p>Разверткой боковой поверхности цилиндра является прямоугольник, одна сторона которого равна высоте цилиндра ($h$), а другая — длине окружности его основания ($2\pi r$).<br>Развертка полной поверхности цилиндра состоит из этого прямоугольника и двух кругов, которые являются основаниями цилиндра.<br><br><strong>Изображение развертки полной поверхности цилиндра:</strong><br><svg width="320" height="220" xmlns="http://www.w3.org/2000/svg" font-family="sans-serif" font-size="16px"><style> .label { font-style: italic; } </style><!-- Прямоугольник (боковая поверхность) --><rect x="60" y="60" width="200" height="100" fill="#f0f0f0" stroke="#333" stroke-width="1.5"/><!-- Верхний круг (основание) --><circle cx="160" cy="60" r="31.83" fill="#d0e0ff" stroke="#333" stroke-width="1.5"/><!-- Нижний круг (основание) --><circle cx="160" cy="160" r="31.83" fill="#d0e0ff" stroke="#333" stroke-width="1.5"/><!-- Обозначения --><!-- Высота h --><line x1="270" y1="60" x2="270" y2="160" stroke="#333" stroke-width="1"/><line x1="267" y1="60" x2="273" y2="60" stroke="#333" stroke-width="1"/><line x1="267" y1="160" x2="273" y2="160" stroke="#333" stroke-width="1"/><text x="278" y="115" class="label">h</text><!-- Длина 2πr --><line x1="60" y1="40" x2="260" y2="40" stroke="#333" stroke-width="1"/><line x1="60" y1="37" x2="60" y2="43" stroke="#333" stroke-width="1"/><line x1="260" y1="37" x2="260" y2="43" stroke="#333" stroke-width="1"/><text x="145" y="32" class="label">2πr</text><!-- Радиус r --><line x1="160" y1="160" x2="191.83" y2="160" stroke="#333" stroke-width="1" stroke-dasharray="3,2"/><text x="170" y="180" class="label">r</text></svg><br><br>Ответ: Разверткой боковой поверхности цилиндра является прямоугольник. Развертка полной поверхности, состоящая из прямоугольника (боковой поверхности) и двух кругов (оснований), изображена выше.</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" => "1144515" "type" => "task" ] "previous" => array:2 [ "refs" => "1144513" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1055} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\BookPage {#1084 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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Вопросы? (с. 97)
Условие. Вопросы? (с. 97)
скриншот условия
ВОПРОСЫ
1. Что принимается за площадь поверхности цилиндра?
2. По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.
3. Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.
Решение. Вопросы? (с. 97)
Решение 2 (rus). Вопросы? (с. 97)
1. Что принимается за площадь поверхности цилиндра?
За площадь полной поверхности цилиндра принимается сумма площадей его боковой поверхности и двух его оснований. Если в задаче говорится просто о «площади поверхности цилиндра» без уточнений, как правило, имеется в виду именно площадь полной поверхности.
Ответ: Сумма площадей боковой поверхности и двух оснований цилиндра.
2. По каким формулам можно найти площади боковой и полной поверхностей цилиндра? Выведите эти формулы.
Пусть дан цилиндр с радиусом основания $r$ и высотой $h$.
Вывод формулы площади боковой поверхности ($S_{бок}$)
Разверткой боковой поверхности цилиндра является прямоугольник. Одна сторона этого прямоугольника равна высоте цилиндра $h$. Другая сторона равна длине окружности основания цилиндра, которая вычисляется по формуле $C = 2\pi r$.
Площадь этого прямоугольника, а следовательно, и площадь боковой поверхности цилиндра, равна произведению его сторон:
$S_{бок} = C \cdot h = 2\pi r \cdot h = 2\pi rh$
Вывод формулы площади полной поверхности ($S_{полн}$)
Площадь полной поверхности цилиндра ($S_{полн}$) равна сумме площади его боковой поверхности ($S_{бок}$) и площадей двух его оснований ($S_{осн}$).
Основанием цилиндра является круг радиусом $r$. Площадь одного такого круга равна:
$S_{осн} = \pi r^2$
Поскольку у цилиндра два основания, их общая площадь составляет $2 \cdot S_{осн} = 2\pi r^2$.
Сложив площадь боковой поверхности и площади двух оснований, получаем формулу для площади полной поверхности:
$S_{полн} = S_{бок} + 2S_{осн} = 2\pi rh + 2\pi r^2$
Эту формулу также можно записать, вынеся за скобки общий множитель $2\pi r$:
$S_{полн} = 2\pi r(h + r)$
Ответ: Площадь боковой поверхности: $S_{бок} = 2\pi rh$. Площадь полной поверхности: $S_{полн} = 2\pi rh + 2\pi r^2 = 2\pi r(h + r)$.
3. Что является разверткой боковой поверхности цилиндра? Изобразите развертку полной поверхности цилиндра.
Разверткой боковой поверхности цилиндра является прямоугольник, одна сторона которого равна высоте цилиндра ($h$), а другая — длине окружности его основания ($2\pi r$).
Развертка полной поверхности цилиндра состоит из этого прямоугольника и двух кругов, которые являются основаниями цилиндра.
Изображение развертки полной поверхности цилиндра:
Ответ: Разверткой боковой поверхности цилиндра является прямоугольник. Развертка полной поверхности, состоящая из прямоугольника (боковой поверхности) и двух кругов (оснований), изображена выше.
Другие задания:
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ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по геометрии за 11 класс, для упражнения Вопросы? расположенного на странице 97 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 Вопросы? (с. 97), авторов: Солтан (Г Н), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.