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Номер 450, страница 121 - 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 5. Объёмы тел. Параграф 1. Объём прямоугольного параллелепипеда - номер 450, страница 121.
App\Models\Task {#1024 // 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" => 744865 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "121" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-450" "field_display_title" => "450" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 744392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы тел" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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" => "116" "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 [ …6] "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 [ …6] "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 {#1036 #items: array:2 [ 0 => App\Models\Term {#1037 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1038 #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 [ …6] "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 [ …6] "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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1041 #items: array:5 [ 0 => App\Models\Term {#1042 #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" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 3753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атанасян" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Левон" "field_patronymic" => "Сергеевич" "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 {#1043 #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" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Бутузов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Валентин" "field_patronymic" => "Фёдорович" "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 {#1044 #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" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "field_surname_rp" => null ] #original: array:12 [ "id" => 4518 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кадомцев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Сергей" "field_patronymic" => "Борисович" "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 => "*" ] } 3 => App\Models\Term {#1045 #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" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "field_surname_rp" => null ] #original: array:12 [ "id" => 5494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Позняк" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Эдуард" "field_patronymic" => "Генрихович" "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 => "*" ] } 4 => App\Models\Term {#1046 #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" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6905 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Киселёва" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Людмила" "field_patronymic" => "Сергеевна" "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 {#1047 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #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 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null …1 ] #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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 {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #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: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 {#1058 #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: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 {#1060 #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 [ …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 {#1062 #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 {#1063 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => 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_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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1039} "field_author" => Illuminate\Database\Eloquent\Collection {#1041} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1047} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1048} "field_country" => Illuminate\Database\Eloquent\Collection {#1050} "field_city" => Illuminate\Database\Eloquent\Collection {#1052} "field_series" => Illuminate\Database\Eloquent\Collection {#1054} "field_umk" => Illuminate\Database\Eloquent\Collection {#1055} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1058} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1062} "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 {#1063} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1064} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1066} "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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы тел" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "116" "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1069 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1070 #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" => 744393 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объём прямоугольного параллелепипеда" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071 #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: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" => "116" "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 {#1073 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ 0 => App\Models\Branch {#1075 #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" => 744392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы тел" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "116" "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 {#1078 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1114 …2} ] #original: array:24 [ "id" => 744392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы тел" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_page_start" => "116" "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 {#1078 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1114 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744393 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объём прямоугольного параллелепипеда" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1071} "field_page_start" => "116" "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 {#1073} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1074} ] #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 {#1115 #items: array:5 [ 0 => App\Models\Element {#1116 #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" => 939099 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Edition {#1118 #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 {#1119 …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 {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123 …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 {#1119 …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 {#1121 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1123 …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" => "744865" "type" => "task" ] "text" => "<p> <b>450.</b> Найдите объём прямоугольного параллелепипеда ABCDA₁B₁C₁D₁, если: <span class="long">a) AC₁ = 1 м, </span> <span class="long">∠C₁AC = 45°, </span> <span class="long">∠C₁AB = 60°; </span> <span class="long">б) АС₁ = 24 см, </span> <span class="long">∠C₁AA₁ = 45°, </span> диагональ АС₁ составляет угол в 30° с плоскостью боковой грани. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1579" "height" => 246 "path" => "/media/geometrija_10/atanasjan-u23/0-00/450-1.webp?ts=1739436639" ] ] ] #original: array:7 [ "id" => 939099 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1117} "task" => array:2 [ "refs" => "744865" "type" => "task" ] "text" => "<p> <b>450.</b> Найдите объём прямоугольного параллелепипеда ABCDA₁B₁C₁D₁, если: <span class="long">a) AC₁ = 1 м, </span> <span class="long">∠C₁AC = 45°, </span> <span class="long">∠C₁AB = 60°; </span> <span class="long">б) АС₁ = 24 см, </span> <span class="long">∠C₁AA₁ = 45°, </span> диагональ АС₁ составляет угол в 30° с плоскостью боковой грани. </p>" "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1579" "height" => 246 "path" => "/media/geometrija_10/atanasjan-u23/0-00/450-1.webp?ts=1739436639" ] ] ] #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 {#1124 #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" => 941832 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Edition {#1126 #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 {#1127 …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 {#1129 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1131 …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 {#1127 …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 {#1129 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1131 …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" => "744865" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "450-1.png" "alt" => null "width" => "694" "height" => 3515 "path" => "/media/geometrija_10/atanasjan-u23/2-00/450-1.webp?ts=1739441325" ] 1 => array:5 [ "name" => "450-2.png" "alt" => null "width" => "694" "height" => 3474 "path" => "/media/geometrija_10/atanasjan-u23/2-00/450-2.webp?ts=1739441325" ] ] ] #original: array:6 [ "id" => 941832 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1125} "task" => array:2 [ "refs" => "744865" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "450-1.png" "alt" => null "width" => "694" "height" => 3515 "path" => "/media/geometrija_10/atanasjan-u23/2-00/450-1.webp?ts=1739441325" ] 1 => array:5 [ "name" => "450-2.png" "alt" => null "width" => "694" "height" => 3474 "path" => "/media/geometrija_10/atanasjan-u23/2-00/450-2.webp?ts=1739441325" ] ] ] #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 {#1132 #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" => 941689 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Edition {#1134 #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 {#1135 …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 {#1137 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …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 {#1135 …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 {#1137 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1139 …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" => "744865" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1378" "height" => 1046 "path" => "/media/geometrija_10/atanasjan-u23/3-00/450-1.webp?ts=1739436909" ] ] ] #original: array:6 [ "id" => 941689 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1133} "task" => array:2 [ "refs" => "744865" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1378" "height" => 1046 "path" => "/media/geometrija_10/atanasjan-u23/3-00/450-1.webp?ts=1739436909" ] ] ] #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 {#1140 #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" => 941203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Edition {#1142 #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 {#1143 …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 {#1145 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1147 …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 {#1143 …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 {#1145 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1147 …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" => "744865" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1412" "height" => 1378 "path" => "/media/geometrija_10/atanasjan-u23/4-00/450-1.webp?ts=1739436589" ] ] ] #original: array:6 [ "id" => 941203 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1141} "task" => array:2 [ "refs" => "744865" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "450-1.jpg" "alt" => null "width" => "1412" "height" => 1378 "path" => "/media/geometrija_10/atanasjan-u23/4-00/450-1.webp?ts=1739436589" ] ] ] #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 {#1148 #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" => 1352178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Edition {#1150 #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 {#1151 …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" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_content_source" => null ] #original: 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 {#1151 …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" "field_edition_type" => "solution" "field_root_dir" => "6-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1154 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1155 …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" => "744865" "type" => "task" ] "text" => "<p><strong>а)</strong>Пусть измерения прямоугольного параллелепипеда равны $a = AB$, $b = AD$ и $c = AA_1$. Объем параллелепипеда вычисляется по формуле $V = a \cdot b \cdot c$.Длина диагонали параллелепипеда $AC_1 = 1$ м.Рассмотрим прямоугольный треугольник $\triangle ACC_1$ (ребро $CC_1$ перпендикулярно основанию $ABC$, а значит, и диагонали основания $AC$). По условию $\angle C_1AC = 45^\circ$.Найдем высоту $c = CC_1$ и диагональ основания $AC$:<br>$c = CC_1 = AC_1 \cdot \sin(\angle C_1AC) = 1 \cdot \sin(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.<br>$AC = AC_1 \cdot \cos(\angle C_1AC) = 1 \cdot \cos(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.<br>Квадрат диагонали прямоугольного основания равен сумме квадратов его сторон: $AC^2 = AB^2 + BC^2 = a^2 + b^2$.Следовательно, $a^2 + b^2 = (\frac{\sqrt{2}}{2})^2 = \frac{2}{4} = \frac{1}{2}$.</p><p>Теперь используем второе условие: $\angle C_1AB = 60^\circ$. Это угол между диагональю параллелепипеда $AC_1$ и ребром $AB$. Проекция диагонали $AC_1$ на прямую, содержащую ребро $AB$, есть само ребро $AB$.Поэтому $AB = AC_1 \cdot \cos(\angle C_1AB)$.<br>$a = 1 \cdot \cos(60^\circ) = 1 \cdot \frac{1}{2} = \frac{1}{2}$ м.</p><p>Подставим найденное значение $a$ в соотношение $a^2 + b^2 = \frac{1}{2}$, чтобы найти $b$:<br>$(\frac{1}{2})^2 + b^2 = \frac{1}{2}$<br>$\frac{1}{4} + b^2 = \frac{1}{2}$<br>$b^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$<br>$b = \sqrt{\frac{1}{4}} = \frac{1}{2}$ м.</p><p>Таким образом, измерения параллелепипеда: $a = \frac{1}{2}$ м, $b = \frac{1}{2}$ м, $c = \frac{\sqrt{2}}{2}$ м.Вычислим объем:<br>$V = a \cdot b \cdot c = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{8}$ м?.</p><p>Ответ: $\frac{\sqrt{2}}{8}$ м?.</p><p><strong>б)</strong>Пусть измерения параллелепипеда $a = AB$, $b = AD$, $c = AA_1$. Объем $V = abc$.Дана диагональ $AC_1 = 24$ см.Рассмотрим прямоугольный треугольник $\triangle AA_1C_1$ (ребро $AA_1$ перпендикулярно верхней грани, а значит и отрезку $A_1C_1$). По условию $\angle C_1AA_1 = 45^\circ$.Найдем высоту $c = AA_1$ и диагональ основания $AC = A_1C_1$:<br>$c = AA_1 = AC_1 \cdot \cos(\angle C_1AA_1) = 24 \cdot \cos(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.<br>$AC = A_1C_1 = AC_1 \cdot \sin(\angle C_1AA_1) = 24 \cdot \sin(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.<br>Из прямоугольника в основании имеем $AC^2 = a^2 + b^2$, откуда $a^2 + b^2 = (12\sqrt{2})^2 = 144 \cdot 2 = 288$.</p><p>Диагональ $AC_1$ составляет угол в $30^\circ$ с плоскостью боковой грани. Возьмем боковую грань $ADD_1A_1$. Угол между прямой и плоскостью — это угол между прямой и ее проекцией на эту плоскость. Проекцией диагонали $AC_1$ на плоскость $ADD_1A_1$ является отрезок $AD_1$. Значит, $\angle C_1AD_1 = 30^\circ$.Рассмотрим треугольник $\triangle AC_1D_1$. Ребро $C_1D_1$ перпендикулярно грани $ADD_1A_1$, следовательно, $C_1D_1 \perp AD_1$. Таким образом, $\triangle AC_1D_1$ — прямоугольный с прямым углом при вершине $D_1$.В этом треугольнике катет $C_1D_1$ лежит напротив угла в $30^\circ$. Длина этого катета равна длине ребра $AB$, то есть $C_1D_1 = a$.<br>$\sin(\angle C_1AD_1) = \frac{C_1D_1}{AC_1}$<br>$\sin(30^\circ) = \frac{a}{24}$<br>$\frac{1}{2} = \frac{a}{24}$<br>Отсюда находим $a = 12$ см.</p><p>Теперь найдем измерение $b$ из соотношения $a^2 + b^2 = 288$:<br>$12^2 + b^2 = 288$<br>$144 + b^2 = 288$<br>$b^2 = 288 - 144 = 144$<br>$b = \sqrt{144} = 12$ см.</p><p>Итак, измерения параллелепипеда: $a = 12$ см, $b = 12$ см, $c = 12\sqrt{2}$ см.Найдем объем:<br>$V = a \cdot b \cdot c = 12 \cdot 12 \cdot 12\sqrt{2} = 1728\sqrt{2}$ см?.</p><p>Ответ: $1728\sqrt{2}$ см?.</p>" ] #original: array:6 [ "id" => 1352178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149} "task" => array:2 [ "refs" => "744865" "type" => "task" ] "text" => "<p><strong>а)</strong>Пусть измерения прямоугольного параллелепипеда равны $a = AB$, $b = AD$ и $c = AA_1$. Объем параллелепипеда вычисляется по формуле $V = a \cdot b \cdot c$.Длина диагонали параллелепипеда $AC_1 = 1$ м.Рассмотрим прямоугольный треугольник $\triangle ACC_1$ (ребро $CC_1$ перпендикулярно основанию $ABC$, а значит, и диагонали основания $AC$). По условию $\angle C_1AC = 45^\circ$.Найдем высоту $c = CC_1$ и диагональ основания $AC$:<br>$c = CC_1 = AC_1 \cdot \sin(\angle C_1AC) = 1 \cdot \sin(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.<br>$AC = AC_1 \cdot \cos(\angle C_1AC) = 1 \cdot \cos(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.<br>Квадрат диагонали прямоугольного основания равен сумме квадратов его сторон: $AC^2 = AB^2 + BC^2 = a^2 + b^2$.Следовательно, $a^2 + b^2 = (\frac{\sqrt{2}}{2})^2 = \frac{2}{4} = \frac{1}{2}$.</p><p>Теперь используем второе условие: $\angle C_1AB = 60^\circ$. Это угол между диагональю параллелепипеда $AC_1$ и ребром $AB$. Проекция диагонали $AC_1$ на прямую, содержащую ребро $AB$, есть само ребро $AB$.Поэтому $AB = AC_1 \cdot \cos(\angle C_1AB)$.<br>$a = 1 \cdot \cos(60^\circ) = 1 \cdot \frac{1}{2} = \frac{1}{2}$ м.</p><p>Подставим найденное значение $a$ в соотношение $a^2 + b^2 = \frac{1}{2}$, чтобы найти $b$:<br>$(\frac{1}{2})^2 + b^2 = \frac{1}{2}$<br>$\frac{1}{4} + b^2 = \frac{1}{2}$<br>$b^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$<br>$b = \sqrt{\frac{1}{4}} = \frac{1}{2}$ м.</p><p>Таким образом, измерения параллелепипеда: $a = \frac{1}{2}$ м, $b = \frac{1}{2}$ м, $c = \frac{\sqrt{2}}{2}$ м.Вычислим объем:<br>$V = a \cdot b \cdot c = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{8}$ м?.</p><p>Ответ: $\frac{\sqrt{2}}{8}$ м?.</p><p><strong>б)</strong>Пусть измерения параллелепипеда $a = AB$, $b = AD$, $c = AA_1$. Объем $V = abc$.Дана диагональ $AC_1 = 24$ см.Рассмотрим прямоугольный треугольник $\triangle AA_1C_1$ (ребро $AA_1$ перпендикулярно верхней грани, а значит и отрезку $A_1C_1$). По условию $\angle C_1AA_1 = 45^\circ$.Найдем высоту $c = AA_1$ и диагональ основания $AC = A_1C_1$:<br>$c = AA_1 = AC_1 \cdot \cos(\angle C_1AA_1) = 24 \cdot \cos(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.<br>$AC = A_1C_1 = AC_1 \cdot \sin(\angle C_1AA_1) = 24 \cdot \sin(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.<br>Из прямоугольника в основании имеем $AC^2 = a^2 + b^2$, откуда $a^2 + b^2 = (12\sqrt{2})^2 = 144 \cdot 2 = 288$.</p><p>Диагональ $AC_1$ составляет угол в $30^\circ$ с плоскостью боковой грани. Возьмем боковую грань $ADD_1A_1$. Угол между прямой и плоскостью — это угол между прямой и ее проекцией на эту плоскость. Проекцией диагонали $AC_1$ на плоскость $ADD_1A_1$ является отрезок $AD_1$. Значит, $\angle C_1AD_1 = 30^\circ$.Рассмотрим треугольник $\triangle AC_1D_1$. Ребро $C_1D_1$ перпендикулярно грани $ADD_1A_1$, следовательно, $C_1D_1 \perp AD_1$. Таким образом, $\triangle AC_1D_1$ — прямоугольный с прямым углом при вершине $D_1$.В этом треугольнике катет $C_1D_1$ лежит напротив угла в $30^\circ$. Длина этого катета равна длине ребра $AB$, то есть $C_1D_1 = a$.<br>$\sin(\angle C_1AD_1) = \frac{C_1D_1}{AC_1}$<br>$\sin(30^\circ) = \frac{a}{24}$<br>$\frac{1}{2} = \frac{a}{24}$<br>Отсюда находим $a = 12$ см.</p><p>Теперь найдем измерение $b$ из соотношения $a^2 + b^2 = 288$:<br>$12^2 + b^2 = 288$<br>$144 + b^2 = 288$<br>$b^2 = 288 - 144 = 144$<br>$b = \sqrt{144} = 12$ см.</p><p>Итак, измерения параллелепипеда: $a = 12$ см, $b = 12$ см, $c = 12\sqrt{2}$ см.Найдем объем:<br>$V = a \cdot b \cdot c = 12 \cdot 12 \cdot 12\sqrt{2} = 1728\sqrt{2}$ см?.</p><p>Ответ: $1728\sqrt{2}$ см?.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "744866" "type" => "task" ] "previous" => array:2 [ "refs" 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App\Models\Element {#1202 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "933904" "type" => "book_page" ] "previous" => array:2 [ "refs" => "933902" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:10 [ 0 => App\Models\Task {#1211 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false 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=> "933903" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№450 (с. 121)
Условие. №450 (с. 121)
скриншот условия
450. Найдите объём прямоугольного параллелепипеда ABCDA₁B₁C₁D₁, если: a) AC₁ = 1 м, ∠C₁AC = 45°, ∠C₁AB = 60°; б) АС₁ = 24 см, ∠C₁AA₁ = 45°, диагональ АС₁ составляет угол в 30° с плоскостью боковой грани.
Решение 2. №450 (с. 121)
Решение 3. №450 (с. 121)
Решение 4. №450 (с. 121)
Решение 6. №450 (с. 121)
а)Пусть измерения прямоугольного параллелепипеда равны $a = AB$, $b = AD$ и $c = AA_1$. Объем параллелепипеда вычисляется по формуле $V = a \cdot b \cdot c$.Длина диагонали параллелепипеда $AC_1 = 1$ м.Рассмотрим прямоугольный треугольник $\triangle ACC_1$ (ребро $CC_1$ перпендикулярно основанию $ABC$, а значит, и диагонали основания $AC$). По условию $\angle C_1AC = 45^\circ$.Найдем высоту $c = CC_1$ и диагональ основания $AC$:
$c = CC_1 = AC_1 \cdot \sin(\angle C_1AC) = 1 \cdot \sin(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.
$AC = AC_1 \cdot \cos(\angle C_1AC) = 1 \cdot \cos(45^\circ) = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$ м.
Квадрат диагонали прямоугольного основания равен сумме квадратов его сторон: $AC^2 = AB^2 + BC^2 = a^2 + b^2$.Следовательно, $a^2 + b^2 = (\frac{\sqrt{2}}{2})^2 = \frac{2}{4} = \frac{1}{2}$.
Теперь используем второе условие: $\angle C_1AB = 60^\circ$. Это угол между диагональю параллелепипеда $AC_1$ и ребром $AB$. Проекция диагонали $AC_1$ на прямую, содержащую ребро $AB$, есть само ребро $AB$.Поэтому $AB = AC_1 \cdot \cos(\angle C_1AB)$.
$a = 1 \cdot \cos(60^\circ) = 1 \cdot \frac{1}{2} = \frac{1}{2}$ м.
Подставим найденное значение $a$ в соотношение $a^2 + b^2 = \frac{1}{2}$, чтобы найти $b$:
$(\frac{1}{2})^2 + b^2 = \frac{1}{2}$
$\frac{1}{4} + b^2 = \frac{1}{2}$
$b^2 = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$
$b = \sqrt{\frac{1}{4}} = \frac{1}{2}$ м.
Таким образом, измерения параллелепипеда: $a = \frac{1}{2}$ м, $b = \frac{1}{2}$ м, $c = \frac{\sqrt{2}}{2}$ м.Вычислим объем:
$V = a \cdot b \cdot c = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{8}$ м?.
Ответ: $\frac{\sqrt{2}}{8}$ м?.
б)Пусть измерения параллелепипеда $a = AB$, $b = AD$, $c = AA_1$. Объем $V = abc$.Дана диагональ $AC_1 = 24$ см.Рассмотрим прямоугольный треугольник $\triangle AA_1C_1$ (ребро $AA_1$ перпендикулярно верхней грани, а значит и отрезку $A_1C_1$). По условию $\angle C_1AA_1 = 45^\circ$.Найдем высоту $c = AA_1$ и диагональ основания $AC = A_1C_1$:
$c = AA_1 = AC_1 \cdot \cos(\angle C_1AA_1) = 24 \cdot \cos(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.
$AC = A_1C_1 = AC_1 \cdot \sin(\angle C_1AA_1) = 24 \cdot \sin(45^\circ) = 24 \cdot \frac{\sqrt{2}}{2} = 12\sqrt{2}$ см.
Из прямоугольника в основании имеем $AC^2 = a^2 + b^2$, откуда $a^2 + b^2 = (12\sqrt{2})^2 = 144 \cdot 2 = 288$.
Диагональ $AC_1$ составляет угол в $30^\circ$ с плоскостью боковой грани. Возьмем боковую грань $ADD_1A_1$. Угол между прямой и плоскостью — это угол между прямой и ее проекцией на эту плоскость. Проекцией диагонали $AC_1$ на плоскость $ADD_1A_1$ является отрезок $AD_1$. Значит, $\angle C_1AD_1 = 30^\circ$.Рассмотрим треугольник $\triangle AC_1D_1$. Ребро $C_1D_1$ перпендикулярно грани $ADD_1A_1$, следовательно, $C_1D_1 \perp AD_1$. Таким образом, $\triangle AC_1D_1$ — прямоугольный с прямым углом при вершине $D_1$.В этом треугольнике катет $C_1D_1$ лежит напротив угла в $30^\circ$. Длина этого катета равна длине ребра $AB$, то есть $C_1D_1 = a$.
$\sin(\angle C_1AD_1) = \frac{C_1D_1}{AC_1}$
$\sin(30^\circ) = \frac{a}{24}$
$\frac{1}{2} = \frac{a}{24}$
Отсюда находим $a = 12$ см.
Теперь найдем измерение $b$ из соотношения $a^2 + b^2 = 288$:
$12^2 + b^2 = 288$
$144 + b^2 = 288$
$b^2 = 288 - 144 = 144$
$b = \sqrt{144} = 12$ см.
Итак, измерения параллелепипеда: $a = 12$ см, $b = 12$ см, $c = 12\sqrt{2}$ см.Найдем объем:
$V = a \cdot b \cdot c = 12 \cdot 12 \cdot 12\sqrt{2} = 1728\sqrt{2}$ см?.
Ответ: $1728\sqrt{2}$ см?.
Другие задания:
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