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Номер 468, страница 130 - 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 3. Объёмы наклонной призмы, пирамиды и конуса - номер 468, страница 130.
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" => 744883 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "130" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-468" "field_display_title" => "468" "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 {#1079 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #items: array:1 [ 0 => 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: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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1070 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1076 #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 {#1039} "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1120 #items: array:2 [ 0 => App\Models\Branch {#1128 #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 {#1132 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1168 #items: array:1 [ 0 => App\Models\Book {#1131 #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 {#1133 #items: array:1 [ 0 => App\Models\Term {#1130 #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 {#1134 #items: array:2 [ 0 => App\Models\Term {#1135 #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 {#1136 #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 {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #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 {#1139 #items: array:5 [ 0 => App\Models\Term {#1140 #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 {#1141 #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 {#1142 #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 {#1143 #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 {#1144 #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" …8 ] #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 {#1145 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #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 {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #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 {#1152 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Term {#1155 #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 {#1156 #items: array:1 [ 0 => App\Models\Term {#1157 #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 {#1158 #items: array:1 [ 0 => App\Models\Term {#1159 #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 {#1160 #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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #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 {#1133} "field_class" => Illuminate\Database\Eloquent\Collection {#1134} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137} "field_author" => Illuminate\Database\Eloquent\Collection {#1139} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1148} "field_city" => Illuminate\Database\Eloquent\Collection {#1150} "field_series" => Illuminate\Database\Eloquent\Collection {#1152} "field_umk" => Illuminate\Database\Eloquent\Collection {#1153} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1160} "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 {#1161} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1162} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1164} "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 {#1166 #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 {#1132} "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 {#1168} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166} ] #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 {#1167 #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" => 744395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы наклонной призмы, пирамиды и конуса" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1169 #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" => "125" "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 {#1171 #items: array:1 [ 0 => App\Models\Book {#1131} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Объёмы наклонной призмы, пирамиды и конуса" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1172} "field_page_start" => "125" "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 {#1171} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #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 {#1117 #items: array:5 [ 0 => App\Models\Element {#1107 #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" => 939117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1108 #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 {#1106 …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 {#1103 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …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 {#1106 …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 {#1103 …2} "field_process_formula" => "wiris" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1101 …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" => "744883" "type" => "task" ] "text" => """ <p><strong>468.</strong> Фигура, заштрихованная на рисунке 148, вращается вокруг оси Оу. Найдите объём полученного тела.</p><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/468.webp?ts=1745670862" alt="Найти объём полученного тела" loading="lazy" width="489" height="560">\n </figure>\n """ "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1565" "height" => 174 "path" => "/media/geometrija_10/atanasjan-u23/0-00/468-1.webp?ts=1739436659" ] ] ] #original: array:7 [ "id" => 939117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "744883" "type" => "task" ] "text" => """ <p><strong>468.</strong> Фигура, заштрихованная на рисунке 148, вращается вокруг оси Оу. Найдите объём полученного тела.</p><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/468.webp?ts=1745670862" alt="Найти объём полученного тела" loading="lazy" width="489" height="560">\n </figure>\n """ "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1565" "height" => 174 "path" => "/media/geometrija_10/atanasjan-u23/0-00/468-1.webp?ts=1739436659" ] ] ] #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 {#1086 #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" => 941891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Edition {#1083 #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 {#1078 …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 {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1127 …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 {#1078 …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 {#1119 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1127 …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" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.png" "alt" => null "width" => "694" "height" => 1641 "path" => "/media/geometrija_10/atanasjan-u23/2-00/468-1.webp?ts=1739441325" ] ] ] #original: array:6 [ "id" => 941891 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124} "task" => array:2 [ "refs" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.png" "alt" => null "width" => "694" "height" => 1641 "path" => "/media/geometrija_10/atanasjan-u23/2-00/468-1.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 {#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" => 941750 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084 #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" => 2318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 …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 {#1088 …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 {#1089 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …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" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1378" "height" => 501 "path" => "/media/geometrija_10/atanasjan-u23/3-00/468-1.webp?ts=1739436933" ] ] ] #original: array:6 [ "id" => 941750 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1084} "task" => array:2 [ "refs" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1378" "height" => 501 "path" => "/media/geometrija_10/atanasjan-u23/3-00/468-1.webp?ts=1739436933" ] ] ] #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 {#1102 #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" => 941276 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Edition {#1099 #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 {#1098 …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 {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1095 …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 {#1098 …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 {#1096 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1301" "height" => 297 "path" => "/media/geometrija_10/atanasjan-u23/4-00/468-1.webp?ts=1739436611" ] ] ] #original: array:6 [ "id" => 941276 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1092} "task" => array:2 [ "refs" => "744883" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "468-1.jpg" "alt" => null "width" => "1301" "height" => 297 "path" => "/media/geometrija_10/atanasjan-u23/4-00/468-1.webp?ts=1739436611" ] ] ] #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 {#1123 #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" => 1352196 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Edition {#1121 #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 {#1173 …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 {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …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 {#1173 …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 {#1174 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1176 …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" => "744883" "type" => "task" ] "text" => "<p>Задача предполагает нахождение объёма тела, полученного вращением плоской фигуры вокруг одной из координатных осей. Так как на изображении не указан конкретный подпункт, а лишь общая формулировка задачи, приведём решение для всех подпунктов, соответствующих рисунку 148 из учебника.</p><p><strong>а) Фигура ограничена линиями $y = x^2, y = 1, x = 0$; вращение вокруг оси $Oy$.</strong></p><p>Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Oy$, применяется формула $V = \pi \int_{c}^{d} [x(y)]^2 dy$.<br>Сначала выразим $x$ как функцию от $y$ из уравнения параболы $y = x^2$. Поскольку фигура находится в первой четверти ($x \ge 0$), получаем $x = \sqrt{y}$.<br>Пределы интегрирования по оси $y$ заданы условиями: от $y=0$ до $y=1$.<br>Теперь подставим всё в формулу и вычислим интеграл: $V = \pi \int_{0}^{1} (\sqrt{y})^2 dy = \pi \int_{0}^{1} y \, dy$<br>$V = \pi \left[ \frac{y^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \pi \left( \frac{1}{2} - 0 \right) = \frac{\pi}{2}$.</p><p>Ответ: $\frac{\pi}{2}$.</p><p><strong>б) Фигура ограничена линиями $y = x^3, y = 1, x = 0$; вращение вокруг оси $Oy$.</strong></p><p>Используем ту же формулу для вращения вокруг оси $Oy$: $V = \pi \int_{c}^{d} [x(y)]^2 dy$.<br>Из уравнения $y = x^3$ выражаем $x$: $x = \sqrt[3]{y}$.<br>Пределы интегрирования по $y$ — от 0 до 1.<br>Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt[3]{y})^2 dy = \pi \int_{0}^{1} y^{2/3} dy$<br>$V = \pi \left[ \frac{y^{2/3+1}}{2/3+1} \right]_{0}^{1} = \pi \left[ \frac{y^{5/3}}{5/3} \right]_{0}^{1} = \pi \left[ \frac{3}{5}y^{5/3} \right]_{0}^{1} = \pi \left( \frac{3}{5} \cdot 1^{5/3} - \frac{3}{5} \cdot 0^{5/3} \right) = \frac{3\pi}{5}$.</p><p>Ответ: $\frac{3\pi}{5}$.</p><p><strong>в) Фигура ограничена линиями $y = \sqrt{x}, y = 0, x = 1$; вращение вокруг оси $Ox$.</strong></p><p>Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Ox$, применяется формула $V = \pi \int_{a}^{b} [y(x)]^2 dx$.<br>Функция $y(x) = \sqrt{x}$ задана в явном виде. Пределы интегрирования по оси $x$ определяются условиями: от $x=0$ (точка пересечения $y=\sqrt{x}$ и $y=0$) до $x=1$.<br>Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt{x})^2 dx = \pi \int_{0}^{1} x \, dx$<br>$V = \pi \left[ \frac{x^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \frac{\pi}{2}$.</p><p>Ответ: $\frac{\pi}{2}$.</p><p><strong>г) Фигура ограничена линиями $y = x^2, y = x^3$; вращение вокруг оси $Ox$.</strong></p><p>Сначала найдём точки пересечения кривых, чтобы определить пределы интегрирования: $x^2 = x^3 \implies x^3 - x^2 = 0 \implies x^2(x-1) = 0$. Точки пересечения при $x=0$ и $x=1$.<br>На интервале $(0, 1)$ график функции $y = x^2$ расположен выше графика $y = x^3$ (например, при $x=0.5$, имеем $0.5^2=0.25$ и $0.5^3=0.125$, и $0.25 > 0.125$).<br>Объём тела вращения находится методом шайб (колец) как разность объёмов тел вращения, образованных верхней ($y_{верх} = x^2$) и нижней ($y_{низ} = x^3$) кривыми.<br>Формула объёма: $V = \pi \int_{a}^{b} (y_{верх}^2(x) - y_{низ}^2(x)) dx$.<br>$V = \pi \int_{0}^{1} ((x^2)^2 - (x^3)^2) dx = \pi \int_{0}^{1} (x^4 - x^6) dx$<br>$V = \pi \left[ \frac{x^5}{5} - \frac{x^7}{7} \right]_{0}^{1} = \pi \left( (\frac{1^5}{5} - \frac{1^7}{7}) - (\frac{0^5}{5} - \frac{0^7}{7}) \right)$<br>$V = \pi \left( \frac{1}{5} - \frac{1}{7} \right) = \pi \left( \frac{7-5}{35} \right) = \frac{2\pi}{35}$.</p><p>Ответ: $\frac{2\pi}{35}$.</p>" ] #original: array:6 [ "id" => 1352196 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1179} "task" => array:2 [ "refs" => "744883" "type" => "task" ] "text" => "<p>Задача предполагает нахождение объёма тела, полученного вращением плоской фигуры вокруг одной из координатных осей. Так как на изображении не указан конкретный подпункт, а лишь общая формулировка задачи, приведём решение для всех подпунктов, соответствующих рисунку 148 из учебника.</p><p><strong>а) Фигура ограничена линиями $y = x^2, y = 1, x = 0$; вращение вокруг оси $Oy$.</strong></p><p>Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Oy$, применяется формула $V = \pi \int_{c}^{d} [x(y)]^2 dy$.<br>Сначала выразим $x$ как функцию от $y$ из уравнения параболы $y = x^2$. Поскольку фигура находится в первой четверти ($x \ge 0$), получаем $x = \sqrt{y}$.<br>Пределы интегрирования по оси $y$ заданы условиями: от $y=0$ до $y=1$.<br>Теперь подставим всё в формулу и вычислим интеграл: $V = \pi \int_{0}^{1} (\sqrt{y})^2 dy = \pi \int_{0}^{1} y \, dy$<br>$V = \pi \left[ \frac{y^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \pi \left( \frac{1}{2} - 0 \right) = \frac{\pi}{2}$.</p><p>Ответ: $\frac{\pi}{2}$.</p><p><strong>б) Фигура ограничена линиями $y = x^3, y = 1, x = 0$; вращение вокруг оси $Oy$.</strong></p><p>Используем ту же формулу для вращения вокруг оси $Oy$: $V = \pi \int_{c}^{d} [x(y)]^2 dy$.<br>Из уравнения $y = x^3$ выражаем $x$: $x = \sqrt[3]{y}$.<br>Пределы интегрирования по $y$ — от 0 до 1.<br>Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt[3]{y})^2 dy = \pi \int_{0}^{1} y^{2/3} dy$<br>$V = \pi \left[ \frac{y^{2/3+1}}{2/3+1} \right]_{0}^{1} = \pi \left[ \frac{y^{5/3}}{5/3} \right]_{0}^{1} = \pi \left[ \frac{3}{5}y^{5/3} \right]_{0}^{1} = \pi \left( \frac{3}{5} \cdot 1^{5/3} - \frac{3}{5} \cdot 0^{5/3} \right) = \frac{3\pi}{5}$.</p><p>Ответ: $\frac{3\pi}{5}$.</p><p><strong>в) Фигура ограничена линиями $y = \sqrt{x}, y = 0, x = 1$; вращение вокруг оси $Ox$.</strong></p><p>Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Ox$, применяется формула $V = \pi \int_{a}^{b} [y(x)]^2 dx$.<br>Функция $y(x) = \sqrt{x}$ задана в явном виде. Пределы интегрирования по оси $x$ определяются условиями: от $x=0$ (точка пересечения $y=\sqrt{x}$ и $y=0$) до $x=1$.<br>Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt{x})^2 dx = \pi \int_{0}^{1} x \, dx$<br>$V = \pi \left[ \frac{x^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \frac{\pi}{2}$.</p><p>Ответ: $\frac{\pi}{2}$.</p><p><strong>г) Фигура ограничена линиями $y = x^2, y = x^3$; вращение вокруг оси $Ox$.</strong></p><p>Сначала найдём точки пересечения кривых, чтобы определить пределы интегрирования: $x^2 = x^3 \implies x^3 - x^2 = 0 \implies x^2(x-1) = 0$. Точки пересечения при $x=0$ и $x=1$.<br>На интервале $(0, 1)$ график функции $y = x^2$ расположен выше графика $y = x^3$ (например, при $x=0.5$, имеем $0.5^2=0.25$ и $0.5^3=0.125$, и $0.25 > 0.125$).<br>Объём тела вращения находится методом шайб (колец) как разность объёмов тел вращения, образованных верхней ($y_{верх} = x^2$) и нижней ($y_{низ} = x^3$) кривыми.<br>Формула объёма: $V = \pi \int_{a}^{b} (y_{верх}^2(x) - y_{низ}^2(x)) dx$.<br>$V = \pi \int_{0}^{1} ((x^2)^2 - (x^3)^2) dx = \pi \int_{0}^{1} (x^4 - x^6) dx$<br>$V = \pi \left[ \frac{x^5}{5} - \frac{x^7}{7} \right]_{0}^{1} = \pi \left( (\frac{1^5}{5} - \frac{1^7}{7}) - (\frac{0^5}{5} - \frac{0^7}{7}) \right)$<br>$V = \pi \left( \frac{1}{5} - \frac{1}{7} \right) = \pi \left( \frac{7-5}{35} \right) = \frac{2\pi}{35}$.</p><p>Ответ: $\frac{2\pi}{35}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] 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№468 (с. 130)
Условие. №468 (с. 130)
скриншот условия
468. Фигура, заштрихованная на рисунке 148, вращается вокруг оси Оу. Найдите объём полученного тела.
Решение 2. №468 (с. 130)
Решение 3. №468 (с. 130)
Решение 4. №468 (с. 130)
Решение 6. №468 (с. 130)
Задача предполагает нахождение объёма тела, полученного вращением плоской фигуры вокруг одной из координатных осей. Так как на изображении не указан конкретный подпункт, а лишь общая формулировка задачи, приведём решение для всех подпунктов, соответствующих рисунку 148 из учебника.
а) Фигура ограничена линиями $y = x^2, y = 1, x = 0$; вращение вокруг оси $Oy$.
Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Oy$, применяется формула $V = \pi \int_{c}^{d} [x(y)]^2 dy$.
Сначала выразим $x$ как функцию от $y$ из уравнения параболы $y = x^2$. Поскольку фигура находится в первой четверти ($x \ge 0$), получаем $x = \sqrt{y}$.
Пределы интегрирования по оси $y$ заданы условиями: от $y=0$ до $y=1$.
Теперь подставим всё в формулу и вычислим интеграл: $V = \pi \int_{0}^{1} (\sqrt{y})^2 dy = \pi \int_{0}^{1} y \, dy$
$V = \pi \left[ \frac{y^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \pi \left( \frac{1}{2} - 0 \right) = \frac{\pi}{2}$.
Ответ: $\frac{\pi}{2}$.
б) Фигура ограничена линиями $y = x^3, y = 1, x = 0$; вращение вокруг оси $Oy$.
Используем ту же формулу для вращения вокруг оси $Oy$: $V = \pi \int_{c}^{d} [x(y)]^2 dy$.
Из уравнения $y = x^3$ выражаем $x$: $x = \sqrt[3]{y}$.
Пределы интегрирования по $y$ — от 0 до 1.
Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt[3]{y})^2 dy = \pi \int_{0}^{1} y^{2/3} dy$
$V = \pi \left[ \frac{y^{2/3+1}}{2/3+1} \right]_{0}^{1} = \pi \left[ \frac{y^{5/3}}{5/3} \right]_{0}^{1} = \pi \left[ \frac{3}{5}y^{5/3} \right]_{0}^{1} = \pi \left( \frac{3}{5} \cdot 1^{5/3} - \frac{3}{5} \cdot 0^{5/3} \right) = \frac{3\pi}{5}$.
Ответ: $\frac{3\pi}{5}$.
в) Фигура ограничена линиями $y = \sqrt{x}, y = 0, x = 1$; вращение вокруг оси $Ox$.
Для вычисления объёма тела, полученного вращением фигуры вокруг оси $Ox$, применяется формула $V = \pi \int_{a}^{b} [y(x)]^2 dx$.
Функция $y(x) = \sqrt{x}$ задана в явном виде. Пределы интегрирования по оси $x$ определяются условиями: от $x=0$ (точка пересечения $y=\sqrt{x}$ и $y=0$) до $x=1$.
Подставляем в формулу и вычисляем: $V = \pi \int_{0}^{1} (\sqrt{x})^2 dx = \pi \int_{0}^{1} x \, dx$
$V = \pi \left[ \frac{x^2}{2} \right]_{0}^{1} = \pi \left( \frac{1^2}{2} - \frac{0^2}{2} \right) = \frac{\pi}{2}$.
Ответ: $\frac{\pi}{2}$.
г) Фигура ограничена линиями $y = x^2, y = x^3$; вращение вокруг оси $Ox$.
Сначала найдём точки пересечения кривых, чтобы определить пределы интегрирования: $x^2 = x^3 \implies x^3 - x^2 = 0 \implies x^2(x-1) = 0$. Точки пересечения при $x=0$ и $x=1$.
На интервале $(0, 1)$ график функции $y = x^2$ расположен выше графика $y = x^3$ (например, при $x=0.5$, имеем $0.5^2=0.25$ и $0.5^3=0.125$, и $0.25 > 0.125$).
Объём тела вращения находится методом шайб (колец) как разность объёмов тел вращения, образованных верхней ($y_{верх} = x^2$) и нижней ($y_{низ} = x^3$) кривыми.
Формула объёма: $V = \pi \int_{a}^{b} (y_{верх}^2(x) - y_{низ}^2(x)) dx$.
$V = \pi \int_{0}^{1} ((x^2)^2 - (x^3)^2) dx = \pi \int_{0}^{1} (x^4 - x^6) dx$
$V = \pi \left[ \frac{x^5}{5} - \frac{x^7}{7} \right]_{0}^{1} = \pi \left( (\frac{1^5}{5} - \frac{1^7}{7}) - (\frac{0^5}{5} - \frac{0^7}{7}) \right)$
$V = \pi \left( \frac{1}{5} - \frac{1}{7} \right) = \pi \left( \frac{7-5}{35} \right) = \frac{2\pi}{35}$.
Ответ: $\frac{2\pi}{35}$.
Другие задания:
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