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Номер 436, страница 115 - 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 4. Цилиндр, конус и шар. Параграф 3. Сфера, дополнительные задачи - номер 436, страница 115.
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" => 744851 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "115" "field_page_end" => null "field_url" => "/10-klass/geometrija/atanasjan-uchebnik/00-436" "field_display_title" => "436" "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" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "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" => "89" "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" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "89" "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 {#1163 #items: array:2 [ 0 => App\Models\Branch {#1077 #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" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Term {#1078 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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" => "89" "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 {#1081 #items: array:1 [ 0 => App\Models\Book {#1082 #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 {#1083 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1111 …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 {#1112 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1115 …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 {#1083 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1111 …2} …30 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744386 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Цилиндр, конус и шар" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1080} "field_page_start" => "89" "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 {#1081} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #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 {#1118 #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" => 744391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сфера, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #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 [ …7] #original: array:7 [ …7] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_page_start" => "112" "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 {#1121 #items: array:1 [ 0 => App\Models\Book {#1122 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Branch {#1158 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 744391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Сфера, дополнительные задачи" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1119} "field_page_start" => "112" "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 {#1121} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157} ] #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 {#1164 #items: array:6 [ 0 => App\Models\Element {#1178 #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" => 939085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1179 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "text" => """ <p><strong>436.</strong> В конус с углом φ при вершине осевого сечения и радиусом основания r вписана сфера радиуса R (т. е. сфера касается основания конуса и каждой его образующей, рис. 129, б). Найдите: а) r, если известны R и φ; б) R, если известны r и φ; в) φ, если <span class="long">R = 1 см, </span><span class="long">r = <span class="squareroot">3</span> см.</span></p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/436.webp?ts=1745619226" alt="Сферы вписаны в конус и цилиндр" loading="lazy" width="841" height="665">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1559" "height" => 253 "path" => "/media/geometrija_10/atanasjan-u23/0-00/436-1.webp?ts=1739436639" ] 1 => array:5 [ "name" => "436-2.jpg" "alt" => null "width" => "809" "height" => 704 "path" => "/media/geometrija_10/atanasjan-u23/0-00/436-2.webp?ts=1739436639" ] ] ] #original: array:7 [ "id" => 939085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "text" => """ <p><strong>436.</strong> В конус с углом φ при вершине осевого сечения и радиусом основания r вписана сфера радиуса R (т. е. сфера касается основания конуса и каждой его образующей, рис. 129, б). Найдите: а) r, если известны R и φ; б) R, если известны r и φ; в) φ, если <span class="long">R = 1 см, </span><span class="long">r = <span class="squareroot">3</span> см.</span></p><figure><figure>\n <img src="/media/geometrija_10/atanasjan-u23/0-00/436.webp?ts=1745619226" alt="Сферы вписаны в конус и цилиндр" loading="lazy" width="841" height="665">\n </figure>\n <figcaption> </figcaption></figure> """ "img" => array:2 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1559" "height" => 253 "path" => "/media/geometrija_10/atanasjan-u23/0-00/436-1.webp?ts=1739436639" ] 1 => array:5 [ "name" => "436-2.jpg" "alt" => null "width" => "809" "height" => 704 "path" => "/media/geometrija_10/atanasjan-u23/0-00/436-2.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 {#1209 #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" => 941775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1210 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "436-1.png" "alt" => null "width" => "694" "height" => 3134 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-1.webp?ts=1739441295" ] 1 => array:5 [ "name" => "436-2.png" "alt" => null "width" => "694" "height" => 3088 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-2.webp?ts=1739441295" ] 2 => array:5 [ "name" => "436-3.png" "alt" => null "width" => "694" "height" => 3235 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-3.webp?ts=1739441295" ] ] ] #original: array:6 [ "id" => 941775 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "436-1.png" "alt" => null "width" => "694" "height" => 3134 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-1.webp?ts=1739441295" ] 1 => array:5 [ "name" => "436-2.png" "alt" => null "width" => "694" "height" => 3088 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-2.webp?ts=1739441295" ] 2 => array:5 [ "name" => "436-3.png" "alt" => null "width" => "694" "height" => 3235 "path" => "/media/geometrija_10/atanasjan-u23/2-00/436-3.webp?ts=1739441295" ] ] ] #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 {#1201 #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" => 941625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Edition {#1202 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1018" "height" => 648 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-1.webp?ts=1739436909" ] 1 => array:5 [ "name" => "436-2.jpg" "alt" => null "width" => "230" "height" => 412 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-2.webp?ts=1739436909" ] 2 => array:5 [ "name" => "436-3.jpg" "alt" => null "width" => "1321" "height" => 394 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-3.webp?ts=1739436909" ] ] ] #original: array:6 [ "id" => 941625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1018" "height" => 648 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-1.webp?ts=1739436909" ] 1 => array:5 [ "name" => "436-2.jpg" "alt" => null "width" => "230" "height" => 412 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-2.webp?ts=1739436909" ] 2 => array:5 [ "name" => "436-3.jpg" "alt" => null "width" => "1321" "height" => 394 "path" => "/media/geometrija_10/atanasjan-u23/3-00/436-3.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 {#1185 #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" => 941116 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1186 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1395" "height" => 911 "path" => "/media/geometrija_10/atanasjan-u23/4-00/436-1.webp?ts=1739436566" ] ] ] #original: array:6 [ "id" => 941116 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "1395" "height" => 911 "path" => "/media/geometrija_10/atanasjan-u23/4-00/436-1.webp?ts=1739436566" ] ] ] #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 {#1193 #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" => 941489 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "437" "height" => 564 "path" => "/media/geometrija_10/atanasjan-u23/5-00/436-1.webp?ts=1739436631" ] ] ] #original: array:6 [ "id" => 941489 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "436-1.jpg" "alt" => null "width" => "437" "height" => 564 "path" => "/media/geometrija_10/atanasjan-u23/5-00/436-1.webp?ts=1739436631" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1217 #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" => 1352164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ 0 => App\Models\Edition {#1218 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "744851" "type" => "task" ] "text" => "<p>Для решения задачи рассмотрим осевое сечение конуса и вписанной в него сферы. Сечением конуса является равнобедренный треугольник с углом $\phi$ при вершине и основанием, равным диаметру основания конуса $2r$. Сечением сферы является большая окружность радиуса $R$, которая вписана в этот равнобедренный треугольник.</p><p>Пусть $P$ - вершина конуса, $O$ - центр его основания, $A$ - точка на окружности основания. Тогда $\triangle POA$ - прямоугольный треугольник, где $PO$ - высота конуса, $OA = r$ - радиус основания, и $\angle APO = \phi/2$.</p><p>Центр вписанной сферы $C$ лежит на высоте $PO$. Расстояние от центра $C$ до основания конуса равно радиусу сферы $R$, то есть $CO = R$. Центр вписанной в треугольник окружности является точкой пересечения его биссектрис. Следовательно, отрезок $AC$ является биссектрисой угла $\angle PAO$.</p><p>В прямоугольном треугольнике $POA$ сумма острых углов равна $90^\circ$, поэтому $\angle PAO = 90^\circ - \angle APO = 90^\circ - \phi/2$.</p><p>Так как $AC$ - биссектриса угла $\angle PAO$, то $\angle CAO = \frac{1}{2}\angle PAO = \frac{1}{2}(90^\circ - \phi/2) = 45^\circ - \phi/4$.</p><p>Рассмотрим прямоугольный треугольник $COA$. В нем катеты $CO = R$ и $OA = r$. Из определения тангенса угла $\angle CAO$ имеем:</p><p>$\tan(\angle CAO) = \frac{CO}{OA} = \frac{R}{r}$</p><p>Подставив выражение для угла $\angle CAO$, получаем основное соотношение, связывающее $r$, $R$ и $\phi$:</p><p>$\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$</p><p>Используя это соотношение, решим все пункты задачи.</p><p><strong>а)</strong> Найти $r$, если известны $R$ и $\phi$.</p><p>Из основного соотношения выразим $r$:</p><p>$r = \frac{R}{\tan(45^\circ - \frac{\phi}{4})}$</p><p>Используя свойство $\cot(x) = \frac{1}{\tan(x)}$, можно записать ответ в другом виде:</p><p>$r = R \cdot \cot(45^\circ - \frac{\phi}{4})$</p><p>Ответ: $r = R \cdot \cot(45^\circ - \frac{\phi}{4})$.</p><p><strong>б)</strong> Найти $R$, если известны $r$ и $\phi$.</p><p>Из основного соотношения $\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$ выразим $R$:</p><p>$R = r \cdot \tan(45^\circ - \frac{\phi}{4})$</p><p>Ответ: $R = r \cdot \tan(45^\circ - \frac{\phi}{4})$.</p><p><strong>в)</strong> Найти $\phi$, если $R = 1$ см, $r = \sqrt{3}$ см.</p><p>Подставим известные значения $R=1$ и $r=\sqrt{3}$ в основное соотношение:</p><p>$\tan(45^\circ - \frac{\phi}{4}) = \frac{1}{\sqrt{3}}$</p><p>Значение тангенса, равное $\frac{1}{\sqrt{3}}$, соответствует углу $30^\circ$. Так как в геометрической задаче угол $\phi$ должен быть в пределах $(0^\circ, 180^\circ)$, то угол $45^\circ - \phi/4$ должен быть в пределах $(0^\circ, 45^\circ)$, поэтому решение единственное.</p><p>$45^\circ - \frac{\phi}{4} = 30^\circ$</p><p>Выразим $\phi$:</p><p>$\frac{\phi}{4} = 45^\circ - 30^\circ$</p><p>$\frac{\phi}{4} = 15^\circ$</p><p>$\phi = 4 \cdot 15^\circ = 60^\circ$</p><p>Ответ: $\phi = 60^\circ$.</p>" ] #original: array:6 [ "id" => 1352164 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1227} "task" => array:2 [ "refs" => "744851" "type" => "task" ] "text" => "<p>Для решения задачи рассмотрим осевое сечение конуса и вписанной в него сферы. Сечением конуса является равнобедренный треугольник с углом $\phi$ при вершине и основанием, равным диаметру основания конуса $2r$. Сечением сферы является большая окружность радиуса $R$, которая вписана в этот равнобедренный треугольник.</p><p>Пусть $P$ - вершина конуса, $O$ - центр его основания, $A$ - точка на окружности основания. Тогда $\triangle POA$ - прямоугольный треугольник, где $PO$ - высота конуса, $OA = r$ - радиус основания, и $\angle APO = \phi/2$.</p><p>Центр вписанной сферы $C$ лежит на высоте $PO$. Расстояние от центра $C$ до основания конуса равно радиусу сферы $R$, то есть $CO = R$. Центр вписанной в треугольник окружности является точкой пересечения его биссектрис. Следовательно, отрезок $AC$ является биссектрисой угла $\angle PAO$.</p><p>В прямоугольном треугольнике $POA$ сумма острых углов равна $90^\circ$, поэтому $\angle PAO = 90^\circ - \angle APO = 90^\circ - \phi/2$.</p><p>Так как $AC$ - биссектриса угла $\angle PAO$, то $\angle CAO = \frac{1}{2}\angle PAO = \frac{1}{2}(90^\circ - \phi/2) = 45^\circ - \phi/4$.</p><p>Рассмотрим прямоугольный треугольник $COA$. В нем катеты $CO = R$ и $OA = r$. Из определения тангенса угла $\angle CAO$ имеем:</p><p>$\tan(\angle CAO) = \frac{CO}{OA} = \frac{R}{r}$</p><p>Подставив выражение для угла $\angle CAO$, получаем основное соотношение, связывающее $r$, $R$ и $\phi$:</p><p>$\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$</p><p>Используя это соотношение, решим все пункты задачи.</p><p><strong>а)</strong> Найти $r$, если известны $R$ и $\phi$.</p><p>Из основного соотношения выразим $r$:</p><p>$r = \frac{R}{\tan(45^\circ - \frac{\phi}{4})}$</p><p>Используя свойство $\cot(x) = \frac{1}{\tan(x)}$, можно записать ответ в другом виде:</p><p>$r = R \cdot \cot(45^\circ - \frac{\phi}{4})$</p><p>Ответ: $r = R \cdot \cot(45^\circ - \frac{\phi}{4})$.</p><p><strong>б)</strong> Найти $R$, если известны $r$ и $\phi$.</p><p>Из основного соотношения $\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$ выразим $R$:</p><p>$R = r \cdot \tan(45^\circ - \frac{\phi}{4})$</p><p>Ответ: $R = r \cdot \tan(45^\circ - \frac{\phi}{4})$.</p><p><strong>в)</strong> Найти $\phi$, если $R = 1$ см, $r = \sqrt{3}$ см.</p><p>Подставим известные значения $R=1$ и $r=\sqrt{3}$ в основное соотношение:</p><p>$\tan(45^\circ - \frac{\phi}{4}) = \frac{1}{\sqrt{3}}$</p><p>Значение тангенса, равное $\frac{1}{\sqrt{3}}$, соответствует углу $30^\circ$. Так как в геометрической задаче угол $\phi$ должен быть в пределах $(0^\circ, 180^\circ)$, то угол $45^\circ - \phi/4$ должен быть в пределах $(0^\circ, 45^\circ)$, поэтому решение единственное.</p><p>$45^\circ - \frac{\phi}{4} = 30^\circ$</p><p>Выразим $\phi$:</p><p>$\frac{\phi}{4} = 45^\circ - 30^\circ$</p><p>$\frac{\phi}{4} = 15^\circ$</p><p>$\phi = 4 \cdot 15^\circ = 60^\circ$</p><p>Ответ: $\phi = 60^\circ$.</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" => "744852" "type" => "task" ] "previous" => array:2 [ "refs" => "744850" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Book {#1176 #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 {#1165 #items: array:1 [ 0 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:2 [ 0 => App\Models\Term {#1174 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1172 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171 #items: array:1 [ 0 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:5 [ 0 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1170 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1168 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Term {#1225 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Term {#1226 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229 #items: array:1 [ 0 => App\Models\Term {#1230 #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 {#1231 #items: array:1 [ 0 => App\Models\Term {#1232 #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 {#1233 #items: array:1 [ 0 => App\Models\Term {#1234 #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 {#1235 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1236 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:1 [ 0 => App\Models\Term {#1238 #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 {#1239 #items: array:1 [ 0 => App\Models\Term {#1240 #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 {#1241 #items: array:1 [ 0 => App\Models\Term {#1242 #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 {#1243 #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 {#1244 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1245 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1246 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/geometrija/atanasjan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ 0 => App\Models\Term {#1248 #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 {#1165} "field_class" => Illuminate\Database\Eloquent\Collection {#1175} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1171} "field_author" => Illuminate\Database\Eloquent\Collection {#1169} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1228} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1229} "field_country" => Illuminate\Database\Eloquent\Collection {#1231} "field_city" => Illuminate\Database\Eloquent\Collection {#1233} "field_series" => Illuminate\Database\Eloquent\Collection {#1235} "field_umk" => Illuminate\Database\Eloquent\Collection {#1236} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1237} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1239} "field_publication_number" => "11" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1241} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1243} "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 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№436 (с. 115)
Условие. №436 (с. 115)
скриншот условия
436. В конус с углом φ при вершине осевого сечения и радиусом основания r вписана сфера радиуса R (т. е. сфера касается основания конуса и каждой его образующей, рис. 129, б). Найдите: а) r, если известны R и φ; б) R, если известны r и φ; в) φ, если R = 1 см, r = 3 см.
Решение 2. №436 (с. 115)
Решение 3. №436 (с. 115)
Решение 4. №436 (с. 115)
Решение 5. №436 (с. 115)
Решение 6. №436 (с. 115)
Для решения задачи рассмотрим осевое сечение конуса и вписанной в него сферы. Сечением конуса является равнобедренный треугольник с углом $\phi$ при вершине и основанием, равным диаметру основания конуса $2r$. Сечением сферы является большая окружность радиуса $R$, которая вписана в этот равнобедренный треугольник.
Пусть $P$ - вершина конуса, $O$ - центр его основания, $A$ - точка на окружности основания. Тогда $\triangle POA$ - прямоугольный треугольник, где $PO$ - высота конуса, $OA = r$ - радиус основания, и $\angle APO = \phi/2$.
Центр вписанной сферы $C$ лежит на высоте $PO$. Расстояние от центра $C$ до основания конуса равно радиусу сферы $R$, то есть $CO = R$. Центр вписанной в треугольник окружности является точкой пересечения его биссектрис. Следовательно, отрезок $AC$ является биссектрисой угла $\angle PAO$.
В прямоугольном треугольнике $POA$ сумма острых углов равна $90^\circ$, поэтому $\angle PAO = 90^\circ - \angle APO = 90^\circ - \phi/2$.
Так как $AC$ - биссектриса угла $\angle PAO$, то $\angle CAO = \frac{1}{2}\angle PAO = \frac{1}{2}(90^\circ - \phi/2) = 45^\circ - \phi/4$.
Рассмотрим прямоугольный треугольник $COA$. В нем катеты $CO = R$ и $OA = r$. Из определения тангенса угла $\angle CAO$ имеем:
$\tan(\angle CAO) = \frac{CO}{OA} = \frac{R}{r}$
Подставив выражение для угла $\angle CAO$, получаем основное соотношение, связывающее $r$, $R$ и $\phi$:
$\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$
Используя это соотношение, решим все пункты задачи.
а) Найти $r$, если известны $R$ и $\phi$.
Из основного соотношения выразим $r$:
$r = \frac{R}{\tan(45^\circ - \frac{\phi}{4})}$
Используя свойство $\cot(x) = \frac{1}{\tan(x)}$, можно записать ответ в другом виде:
$r = R \cdot \cot(45^\circ - \frac{\phi}{4})$
Ответ: $r = R \cdot \cot(45^\circ - \frac{\phi}{4})$.
б) Найти $R$, если известны $r$ и $\phi$.
Из основного соотношения $\tan(45^\circ - \frac{\phi}{4}) = \frac{R}{r}$ выразим $R$:
$R = r \cdot \tan(45^\circ - \frac{\phi}{4})$
Ответ: $R = r \cdot \tan(45^\circ - \frac{\phi}{4})$.
в) Найти $\phi$, если $R = 1$ см, $r = \sqrt{3}$ см.
Подставим известные значения $R=1$ и $r=\sqrt{3}$ в основное соотношение:
$\tan(45^\circ - \frac{\phi}{4}) = \frac{1}{\sqrt{3}}$
Значение тангенса, равное $\frac{1}{\sqrt{3}}$, соответствует углу $30^\circ$. Так как в геометрической задаче угол $\phi$ должен быть в пределах $(0^\circ, 180^\circ)$, то угол $45^\circ - \phi/4$ должен быть в пределах $(0^\circ, 45^\circ)$, поэтому решение единственное.
$45^\circ - \frac{\phi}{4} = 30^\circ$
Выразим $\phi$:
$\frac{\phi}{4} = 45^\circ - 30^\circ$
$\frac{\phi}{4} = 15^\circ$
$\phi = 4 \cdot 15^\circ = 60^\circ$
Ответ: $\phi = 60^\circ$.
Другие задания:
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