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Консультация № 191292
22.08.2017, 10:34
0.00 руб.
1
1
1
Здравствуйте! Уважаемые эксперты помогите с решением:
Найти стороны прямоугольника наибольшей площади, который
можно вписать в эллипс.
x2/25+y2/9=1
Найти стороны прямоугольника наибольшей площади, который
можно вписать в эллипс.
x2/25+y2/9=1
Прикрепленные файлы:
Обсуждение
22.08.2017, 20:55
общий
это ответ
Здравствуйте, Руслан!
Соображения симметрии подсказывают, что наибольшую площадь будет иметь прямоугольник, стороны которого параллельны осям эллипса.
Перепишем уравнение эллипса так:
Отсюда получим, что
Если стороны прямоугольника, вписанного в эллипс, параллельны его осям, то они делятся этими осями пополам. Примем за
координаты правой верхней вершины прямоугольника. При этом площадь прямоугольника выражается формулой
Рассматривая
как функцию 
найдём точку её максимума. Имеем
то есть максимум достигается при 
При этом
Значит, стороны прямоугольника наибольшей площади составляют
и 
Соображения симметрии подсказывают, что наибольшую площадь будет иметь прямоугольник, стороны которого параллельны осям эллипса.
Перепишем уравнение эллипса так:
Отсюда получим, что
Если стороны прямоугольника, вписанного в эллипс, параллельны его осям, то они делятся этими осями пополам. Примем за
координаты правой верхней вершины прямоугольника. При этом площадь прямоугольника выражается формулойРассматривая
как функцию найдём точку её максимума. Имеемто есть максимум
достигается при При этомЗначит, стороны прямоугольника наибольшей площади составляют
и
5
Спасибо!
Об авторе:
Facta loquuntur.
Facta loquuntur.
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