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Номер 229, страница 71 - гдз по геометрии 10-11 класс учебник Атанасян, Бутузов
Авторы: Атанасян Л. С., Бутузов В. Ф., Кадомцев С. Б., Позняк Э. Г., Киселёва Л. С.
Тип: Учебник
Издательство: Просвещение
Год издания: 2019 - 2026
Уровень обучения: базовый и углублённый
Цвет обложки: коричневый с ромбами
ISBN: 978-5-09-103606-0 (2023)
Допущено Министерством просвещения Российской Федерации
Математика: алгебра и начала математического анализа, геометрия
Популярные ГДЗ в 10 классе
Глава 3. Многогранники. Параграф 1. Понятие многогранника. Призма - номер 229, страница 71.
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" => 744644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "71" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-229" "field_display_title" => "229" "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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "63" "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 {#1086 #items: array:1 [ 0 => App\Models\Book {#1049 #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" => 36 "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #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 [ …10] #original: array:10 [ …10] #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_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:2 [ 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] } 1 => App\Models\Term {#1054 #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_publisher" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 #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_author" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:5 [ 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:12 [ …12] #original: array:12 [ …12] #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\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:12 [ …12] #original: array:12 [ …12] #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\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:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => 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:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1062 #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 [ …12] #original: array:12 [ …12] #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_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => 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:10 [ …10] #original: array:10 [ …10] #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_country" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ 0 => App\Models\Term {#1067 #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_city" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ 0 => App\Models\Term {#1069 #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_series" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Term {#1073 #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_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Term {#1075 #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 [ …10] #original: array:10 [ …10] #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_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => 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: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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1083 #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_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #original: array:50 [ "id" => 36 "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 {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055} "field_author" => Illuminate\Database\Eloquent\Collection {#1057} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1063} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_country" => Illuminate\Database\Eloquent\Collection {#1066} "field_city" => Illuminate\Database\Eloquent\Collection {#1068} "field_series" => Illuminate\Database\Eloquent\Collection {#1070} "field_umk" => Illuminate\Database\Eloquent\Collection {#1071} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1072} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1074} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1076} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1078} "field_under_the_edition_degree" => null "field_cover_description" => "с ромбами" "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для общеобразовательных организаций" "field_allowed" => "Допущено Министерством просвещения Российской Федерации" "field_reserve_field" => "Математика: алгебра и начала математического анализа, геометрия" "field_link_to_source" => null "field_tasks_count" => "1147" "field_priority" => "12" "field_default_folder" => "/geometrija_10/atanasjan-u23/" "field_isbn" => "978-5-09-103606-0 (2023)" "field_cover" => array:1 [ 0 => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_10/atanasjan-u23/covers/cover1.webp?ts=1739363242" "alt" => "" "width" => "1200" "height" => "1661" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1079} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1080} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1081} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1082} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 543 "class_subject" => 392 ] ] #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 {#1084 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "63" "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 {#1086} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1084} ] #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 {#1121 #items: array:2 [ 0 => App\Models\Branch {#1129 #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" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "63" "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 {#1131 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Многогранники" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1130} "field_page_start" => "63" "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 {#1131} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1132} ] #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 {#1133 #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" => 744381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие многогранника. Призма" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "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_page_start" => "63" "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 {#1136 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Понятие многогранника. Призма" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1137} "field_page_start" => "63" "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 {#1136} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1135} ] #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 {#1038 #items: array:6 [ 0 => App\Models\Element {#1109 #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" => 938878 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Edition {#1106 #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" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1105 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "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" => "744644" "type" => "task" ] "text" => "<p> <b>229.</b> В правильной <i>n</i>-угольной призме сторона основания равна <i>а</i> и высота равна <i>h</i>. Вычислите площади боковой и полной поверхности призмы, если: <span class="long">а) n = 3, </span> <span class="long">а = 10 см, </span> <span class="long">h = 15 см; </span> <span class="long">б) n = 4, </span> <span class="long">а = 12 дм, </span> <span class="long">h = 8 дм; </span> <span class="long">в) n = 6, </span> <span class="long">a = 23 см, </span> <span class="long"> h = 5 дм; </span> <span class="long">г) n = 5, </span> <span class="long">а = 0,4 м, </span> <span class="long"> h = 10 см.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1545" "height" => 253 "path" => "/media/geometrija_10/atanasjan-u23/0-00/229-1.webp?ts=1739436471" ] ] ] #original: array:7 [ "id" => 938878 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1098} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "text" => "<p> <b>229.</b> В правильной <i>n</i>-угольной призме сторона основания равна <i>а</i> и высота равна <i>h</i>. Вычислите площади боковой и полной поверхности призмы, если: <span class="long">а) n = 3, </span> <span class="long">а = 10 см, </span> <span class="long">h = 15 см; </span> <span class="long">б) n = 4, </span> <span class="long">а = 12 дм, </span> <span class="long">h = 8 дм; </span> <span class="long">в) n = 6, </span> <span class="long">a = 23 см, </span> <span class="long"> h = 5 дм; </span> <span class="long">г) n = 5, </span> <span class="long">а = 0,4 м, </span> <span class="long"> h = 10 см.</span> </p>" "img" => array:1 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1545" "height" => 253 "path" => "/media/geometrija_10/atanasjan-u23/0-00/229-1.webp?ts=1739436471" ] ] ] #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 {#1122 #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" => 940692 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145 #items: array:1 [ 0 => App\Models\Edition {#1125 #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" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1139 …2} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2317 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1139 …2} "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" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "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" => "744644" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "229-1.png" "alt" => null "width" => "694" "height" => 1621 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-1.webp?ts=1739441078" ] 1 => array:5 [ "name" => "229-2.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-2.webp?ts=1739441078" ] 2 => array:5 [ "name" => "229-3.png" "alt" => null "width" => "694" "height" => 1635 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-3.webp?ts=1739441078" ] 3 => array:5 [ "name" => "229-4.png" "alt" => null "width" => "694" "height" => 1626 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-4.webp?ts=1739441078" ] ] ] #original: array:6 [ "id" => 940692 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1145} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "img" => array:4 [ 0 => array:5 [ "name" => "229-1.png" "alt" => null "width" => "694" "height" => 1621 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-1.webp?ts=1739441078" ] 1 => array:5 [ "name" => "229-2.png" "alt" => null "width" => "694" "height" => 1608 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-2.webp?ts=1739441078" ] 2 => array:5 [ "name" => "229-3.png" "alt" => null "width" => "694" "height" => 1635 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-3.webp?ts=1739441078" ] 3 => array:5 [ "name" => "229-4.png" "alt" => null "width" => "694" "height" => 1626 "path" => "/media/geometrija_10/atanasjan-u23/2-00/229-4.webp?ts=1739441078" ] ] ] #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 {#1100 #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" => 940415 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1101 #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" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 …2} "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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1097 …2} "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" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …2} "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" => "744644" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1375" "height" => 662 "path" => "/media/geometrija_10/atanasjan-u23/3-00/229-1.webp?ts=1739436673" ] 1 => array:5 [ "name" => "229-2.jpg" "alt" => null "width" => "1375" "height" => 1980 "path" => "/media/geometrija_10/atanasjan-u23/3-00/229-2.webp?ts=1739436673" ] ] ] #original: array:6 [ "id" => 940415 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1375" "height" => 662 "path" => "/media/geometrija_10/atanasjan-u23/3-00/229-1.webp?ts=1739436673" ] 1 => array:5 [ "name" => "229-2.jpg" "alt" => null "width" => "1375" "height" => 1980 "path" => "/media/geometrija_10/atanasjan-u23/3-00/229-2.webp?ts=1739436673" ] ] ] #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\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" => 940490 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ 0 => App\Models\Edition {#1093 #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" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …2} "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" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1090 …2} "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" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …2} "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" => "744644" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1148" "height" => 626 "path" => "/media/geometrija_10/atanasjan-u23/4-00/229-1.webp?ts=1739436455" ] 1 => array:5 [ "name" => "229-2.jpg" "alt" => null "width" => "1148" "height" => 371 "path" => "/media/geometrija_10/atanasjan-u23/4-00/229-2.webp?ts=1739436455" ] ] ] #original: array:6 [ "id" => 940490 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1034} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "1148" "height" => 626 "path" => "/media/geometrija_10/atanasjan-u23/4-00/229-1.webp?ts=1739436455" ] 1 => array:5 [ "name" => "229-2.jpg" "alt" => null "width" => "1148" "height" => 371 "path" => "/media/geometrija_10/atanasjan-u23/4-00/229-2.webp?ts=1739436455" ] ] ] #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\Element {#1043 #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" => 940665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Edition {#1041 #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" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "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" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_content_source" => null ] #original: array:21 [ "id" => 2320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …2} "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" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …2} "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" => "744644" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "541" "height" => 699 "path" => "/media/geometrija_10/atanasjan-u23/5-00/229-1.webp?ts=1739436496" ] ] ] #original: array:6 [ "id" => 940665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1123} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "229-1.jpg" "alt" => null "width" => "541" "height" => 699 "path" => "/media/geometrija_10/atanasjan-u23/5-00/229-1.webp?ts=1739436496" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1143 #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" => 1351957 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Edition {#1144 #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" => 5120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 6" "field_order" => "7" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1147 …2} "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" => "vadim" …8 ] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744644" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся общими формулами для правильной n-угольной призмы.</p><p>Площадь боковой поверхности ($S_{бок}$) вычисляется как произведение периметра основания ($P_{осн}$) на высоту призмы ($h$). Периметр основания, в свою очередь, равен произведению числа сторон ($n$) на длину стороны ($a$).</p><p>$S_{бок} = P_{осн} \cdot h = (n \cdot a) \cdot h$</p><p>Площадь основания ($S_{осн}$), которое является правильным n-угольником со стороной $a$, вычисляется по формуле:</p><p>$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})}$</p><p>Площадь полной поверхности ($S_{полн}$) равна сумме площади боковой поверхности и удвоенной площади основания.</p><p>$S_{полн} = S_{бок} + 2S_{осн}$</p><p>Рассмотрим каждый случай отдельно.</p><p><strong>а)</strong></p><p>Дано: $n=3$, $a=10$ см, $h=15$ см. Основание призмы — правильный (равносторонний) треугольник.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 3 \cdot 10 \cdot 15 = 450 \text{ см}^2$</p><p>2. Находим площадь основания. Для равностороннего треугольника ($n=3$) формула площади: $S_{осн} = \frac{a^2 \sqrt{3}}{4}$.</p><p>$S_{осн} = \frac{10^2 \sqrt{3}}{4} = \frac{100\sqrt{3}}{4} = 25\sqrt{3} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 450 + 2 \cdot 25\sqrt{3} = (450 + 50\sqrt{3}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 450 \text{ см}^2$, $S_{полн} = (450 + 50\sqrt{3}) \text{ см}^2$.</p><p><strong>б)</strong></p><p>Дано: $n=4$, $a=12$ дм, $h=8$ дм. Основание призмы — квадрат.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 4 \cdot 12 \cdot 8 = 384 \text{ дм}^2$</p><p>2. Находим площадь основания. Для квадрата ($n=4$) формула площади: $S_{осн} = a^2$.</p><p>$S_{осн} = 12^2 = 144 \text{ дм}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 384 + 2 \cdot 144 = 384 + 288 = 672 \text{ дм}^2$</p><p>Ответ: $S_{бок} = 384 \text{ дм}^2$, $S_{полн} = 672 \text{ дм}^2$.</p><p><strong>в)</strong></p><p>Дано: $n=6$, $a=23$ см, $h=5$ дм. Основание призмы — правильный шестиугольник. Сначала приведем все размеры к одной единице измерения, например, к сантиметрам: $h = 5 \text{ дм} = 50 \text{ см}$.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 6 \cdot 23 \cdot 50 = 6900 \text{ см}^2$</p><p>2. Находим площадь основания. Для правильного шестиугольника ($n=6$) формула площади: $S_{осн} = \frac{3a^2 \sqrt{3}}{2}$.</p><p>$S_{осн} = \frac{3 \cdot 23^2 \sqrt{3}}{2} = \frac{3 \cdot 529 \sqrt{3}}{2} = \frac{1587\sqrt{3}}{2} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 6900 + 2 \cdot \frac{1587\sqrt{3}}{2} = (6900 + 1587\sqrt{3}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 6900 \text{ см}^2$, $S_{полн} = (6900 + 1587\sqrt{3}) \text{ см}^2$.</p><p><strong>г)</strong></p><p>Дано: $n=5$, $a=0,4$ м, $h=10$ см. Основание призмы — правильный пятиугольник. Приведем все размеры к сантиметрам: $a = 0,4 \text{ м} = 40 \text{ см}$.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 5 \cdot 40 \cdot 10 = 2000 \text{ см}^2$</p><p>2. Находим площадь основания. Используем общую формулу для правильного пятиугольника ($n=5$):</p><p>$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})} = \frac{5 \cdot 40^2}{4 \tan(\frac{180^\circ}{5})} = \frac{5 \cdot 1600}{4 \tan(36^\circ)} = \frac{2000}{\tan(36^\circ)} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 2000 + 2 \cdot \frac{2000}{\tan(36^\circ)} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 2000 \text{ см}^2$, $S_{полн} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$.</p>" ] #original: array:6 [ "id" => 1351957 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153} "task" => array:2 [ "refs" => "744644" "type" => "task" ] "text" => "<p>Для решения задачи воспользуемся общими формулами для правильной n-угольной призмы.</p><p>Площадь боковой поверхности ($S_{бок}$) вычисляется как произведение периметра основания ($P_{осн}$) на высоту призмы ($h$). Периметр основания, в свою очередь, равен произведению числа сторон ($n$) на длину стороны ($a$).</p><p>$S_{бок} = P_{осн} \cdot h = (n \cdot a) \cdot h$</p><p>Площадь основания ($S_{осн}$), которое является правильным n-угольником со стороной $a$, вычисляется по формуле:</p><p>$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})}$</p><p>Площадь полной поверхности ($S_{полн}$) равна сумме площади боковой поверхности и удвоенной площади основания.</p><p>$S_{полн} = S_{бок} + 2S_{осн}$</p><p>Рассмотрим каждый случай отдельно.</p><p><strong>а)</strong></p><p>Дано: $n=3$, $a=10$ см, $h=15$ см. Основание призмы — правильный (равносторонний) треугольник.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 3 \cdot 10 \cdot 15 = 450 \text{ см}^2$</p><p>2. Находим площадь основания. Для равностороннего треугольника ($n=3$) формула площади: $S_{осн} = \frac{a^2 \sqrt{3}}{4}$.</p><p>$S_{осн} = \frac{10^2 \sqrt{3}}{4} = \frac{100\sqrt{3}}{4} = 25\sqrt{3} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 450 + 2 \cdot 25\sqrt{3} = (450 + 50\sqrt{3}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 450 \text{ см}^2$, $S_{полн} = (450 + 50\sqrt{3}) \text{ см}^2$.</p><p><strong>б)</strong></p><p>Дано: $n=4$, $a=12$ дм, $h=8$ дм. Основание призмы — квадрат.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 4 \cdot 12 \cdot 8 = 384 \text{ дм}^2$</p><p>2. Находим площадь основания. Для квадрата ($n=4$) формула площади: $S_{осн} = a^2$.</p><p>$S_{осн} = 12^2 = 144 \text{ дм}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 384 + 2 \cdot 144 = 384 + 288 = 672 \text{ дм}^2$</p><p>Ответ: $S_{бок} = 384 \text{ дм}^2$, $S_{полн} = 672 \text{ дм}^2$.</p><p><strong>в)</strong></p><p>Дано: $n=6$, $a=23$ см, $h=5$ дм. Основание призмы — правильный шестиугольник. Сначала приведем все размеры к одной единице измерения, например, к сантиметрам: $h = 5 \text{ дм} = 50 \text{ см}$.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 6 \cdot 23 \cdot 50 = 6900 \text{ см}^2$</p><p>2. Находим площадь основания. Для правильного шестиугольника ($n=6$) формула площади: $S_{осн} = \frac{3a^2 \sqrt{3}}{2}$.</p><p>$S_{осн} = \frac{3 \cdot 23^2 \sqrt{3}}{2} = \frac{3 \cdot 529 \sqrt{3}}{2} = \frac{1587\sqrt{3}}{2} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 6900 + 2 \cdot \frac{1587\sqrt{3}}{2} = (6900 + 1587\sqrt{3}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 6900 \text{ см}^2$, $S_{полн} = (6900 + 1587\sqrt{3}) \text{ см}^2$.</p><p><strong>г)</strong></p><p>Дано: $n=5$, $a=0,4$ м, $h=10$ см. Основание призмы — правильный пятиугольник. Приведем все размеры к сантиметрам: $a = 0,4 \text{ м} = 40 \text{ см}$.</p><p>1. Находим площадь боковой поверхности:</p><p>$S_{бок} = n \cdot a \cdot h = 5 \cdot 40 \cdot 10 = 2000 \text{ см}^2$</p><p>2. Находим площадь основания. Используем общую формулу для правильного пятиугольника ($n=5$):</p><p>$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})} = \frac{5 \cdot 40^2}{4 \tan(\frac{180^\circ}{5})} = \frac{5 \cdot 1600}{4 \tan(36^\circ)} = \frac{2000}{\tan(36^\circ)} \text{ см}^2$</p><p>3. Находим площадь полной поверхности:</p><p>$S_{полн} = S_{бок} + 2S_{осн} = 2000 + 2 \cdot \frac{2000}{\tan(36^\circ)} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$</p><p>Ответ: $S_{бок} = 2000 \text{ см}^2$, $S_{полн} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$.</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" => "744645" "type" => "task" ] "previous" => array:2 [ "refs" => "744643" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\BookPage {#1115 #connection: "mysql" #table: 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"mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #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\Task {#1287 #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 [ …24] #original: array:24 [ …24] #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\Task {#1290 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1289 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: 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+incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1331 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1351 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 8 => App\Models\Task {#1371 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 9 => App\Models\Task {#1391 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 10 => App\Models\Task {#1454 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] 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№229 (с. 71)
Условие. №229 (с. 71)
скриншот условия
229. В правильной n-угольной призме сторона основания равна а и высота равна h. Вычислите площади боковой и полной поверхности призмы, если: а) n = 3, а = 10 см, h = 15 см; б) n = 4, а = 12 дм, h = 8 дм; в) n = 6, a = 23 см, h = 5 дм; г) n = 5, а = 0,4 м, h = 10 см.
Решение 2. №229 (с. 71)
Решение 3. №229 (с. 71)
Решение 4. №229 (с. 71)
Решение 5. №229 (с. 71)
Решение 6. №229 (с. 71)
Для решения задачи воспользуемся общими формулами для правильной n-угольной призмы.
Площадь боковой поверхности ($S_{бок}$) вычисляется как произведение периметра основания ($P_{осн}$) на высоту призмы ($h$). Периметр основания, в свою очередь, равен произведению числа сторон ($n$) на длину стороны ($a$).
$S_{бок} = P_{осн} \cdot h = (n \cdot a) \cdot h$
Площадь основания ($S_{осн}$), которое является правильным n-угольником со стороной $a$, вычисляется по формуле:
$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})}$
Площадь полной поверхности ($S_{полн}$) равна сумме площади боковой поверхности и удвоенной площади основания.
$S_{полн} = S_{бок} + 2S_{осн}$
Рассмотрим каждый случай отдельно.
а)
Дано: $n=3$, $a=10$ см, $h=15$ см. Основание призмы — правильный (равносторонний) треугольник.
1. Находим площадь боковой поверхности:
$S_{бок} = n \cdot a \cdot h = 3 \cdot 10 \cdot 15 = 450 \text{ см}^2$
2. Находим площадь основания. Для равностороннего треугольника ($n=3$) формула площади: $S_{осн} = \frac{a^2 \sqrt{3}}{4}$.
$S_{осн} = \frac{10^2 \sqrt{3}}{4} = \frac{100\sqrt{3}}{4} = 25\sqrt{3} \text{ см}^2$
3. Находим площадь полной поверхности:
$S_{полн} = S_{бок} + 2S_{осн} = 450 + 2 \cdot 25\sqrt{3} = (450 + 50\sqrt{3}) \text{ см}^2$
Ответ: $S_{бок} = 450 \text{ см}^2$, $S_{полн} = (450 + 50\sqrt{3}) \text{ см}^2$.
б)
Дано: $n=4$, $a=12$ дм, $h=8$ дм. Основание призмы — квадрат.
1. Находим площадь боковой поверхности:
$S_{бок} = n \cdot a \cdot h = 4 \cdot 12 \cdot 8 = 384 \text{ дм}^2$
2. Находим площадь основания. Для квадрата ($n=4$) формула площади: $S_{осн} = a^2$.
$S_{осн} = 12^2 = 144 \text{ дм}^2$
3. Находим площадь полной поверхности:
$S_{полн} = S_{бок} + 2S_{осн} = 384 + 2 \cdot 144 = 384 + 288 = 672 \text{ дм}^2$
Ответ: $S_{бок} = 384 \text{ дм}^2$, $S_{полн} = 672 \text{ дм}^2$.
в)
Дано: $n=6$, $a=23$ см, $h=5$ дм. Основание призмы — правильный шестиугольник. Сначала приведем все размеры к одной единице измерения, например, к сантиметрам: $h = 5 \text{ дм} = 50 \text{ см}$.
1. Находим площадь боковой поверхности:
$S_{бок} = n \cdot a \cdot h = 6 \cdot 23 \cdot 50 = 6900 \text{ см}^2$
2. Находим площадь основания. Для правильного шестиугольника ($n=6$) формула площади: $S_{осн} = \frac{3a^2 \sqrt{3}}{2}$.
$S_{осн} = \frac{3 \cdot 23^2 \sqrt{3}}{2} = \frac{3 \cdot 529 \sqrt{3}}{2} = \frac{1587\sqrt{3}}{2} \text{ см}^2$
3. Находим площадь полной поверхности:
$S_{полн} = S_{бок} + 2S_{осн} = 6900 + 2 \cdot \frac{1587\sqrt{3}}{2} = (6900 + 1587\sqrt{3}) \text{ см}^2$
Ответ: $S_{бок} = 6900 \text{ см}^2$, $S_{полн} = (6900 + 1587\sqrt{3}) \text{ см}^2$.
г)
Дано: $n=5$, $a=0,4$ м, $h=10$ см. Основание призмы — правильный пятиугольник. Приведем все размеры к сантиметрам: $a = 0,4 \text{ м} = 40 \text{ см}$.
1. Находим площадь боковой поверхности:
$S_{бок} = n \cdot a \cdot h = 5 \cdot 40 \cdot 10 = 2000 \text{ см}^2$
2. Находим площадь основания. Используем общую формулу для правильного пятиугольника ($n=5$):
$S_{осн} = \frac{n a^2}{4 \tan(\frac{180^\circ}{n})} = \frac{5 \cdot 40^2}{4 \tan(\frac{180^\circ}{5})} = \frac{5 \cdot 1600}{4 \tan(36^\circ)} = \frac{2000}{\tan(36^\circ)} \text{ см}^2$
3. Находим площадь полной поверхности:
$S_{полн} = S_{бок} + 2S_{осн} = 2000 + 2 \cdot \frac{2000}{\tan(36^\circ)} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$
Ответ: $S_{бок} = 2000 \text{ см}^2$, $S_{полн} = (2000 + \frac{4000}{\tan(36^\circ)}) \text{ см}^2$.
Другие задания:
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