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Консультация № 197894
02.03.2020, 21:18
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Объем цилиндрической балки длины l, вырезанной из бревна (имеющего форму усеченного конуса) и соосной с ним, равен V=al(l-b)(l-b), a и b - положительные постоянные, зависящие от размеров бревна (длина бревна меньше чем b но меньше чем b/3). При каком значении l объем такой балки будет наибольшим?
Заранее спасибо!!!
Объем цилиндрической балки длины l, вырезанной из бревна (имеющего форму усеченного конуса) и соосной с ним, равен V=al(l-b)(l-b), a и b - положительные постоянные, зависящие от размеров бревна (длина бревна меньше чем b но меньше чем b/3). При каком значении l объем такой балки будет наибольшим?
Заранее спасибо!!!
Обсуждение
03.03.2020, 02:31
общий
Адресаты:
Сначала в ниже-абзаце "Добавить сообщение…" кликните по чёрному треугольничку правее поля "Кому". Из выпадающего списка выберите "Модераторы" и просите модераторов удалить Вашу избыточную дубль-консультацию rfpro.ru/question/197895 . Экспертов наказывают за ответы на мусор-дубли.
Приложите скриншот исходника Вашей задачи, я подозреваю наличие 2х ошибок в Вашей текст-копии.
Приложите скриншот исходника Вашей задачи, я подозреваю наличие 2х ошибок в Вашей текст-копии.
12.03.2020, 18:38
общий
это ответ
Здравствуйте, KSIW2S!
Запишем объём балки как функцию переменной l:
Требуется найти точки максимума этой функции.
Как известно из курса мат.анализа, необходимым (но не достаточным!) условием наличия в точке экстремума (максимума или минимума) функции является равенство нулю производной функции в этой точке. Достаточным условием является неравенство нулю второй производной, в частности, в точке максимума вторая производная строго меньше нуля.
В данном случае для функции V(l) имеем:
Первая производная обращается в ноль при l = b и l = b/3 - это возможные точки экстремума. Для второй производной имеем V"(b) = 2ab > 0 и V"(b/3) = -2ab < 0. Следовательно, l = b - точка минимума, а l = b/3 - точка максимума.
Итак, при l = b/3 объём балки будет наибольшим и равным 4ab[sup]3[/sup]/27.
Запишем объём балки как функцию переменной l:
Требуется найти точки максимума этой функции.
Как известно из курса мат.анализа, необходимым (но не достаточным!) условием наличия в точке экстремума (максимума или минимума) функции является равенство нулю производной функции в этой точке. Достаточным условием является неравенство нулю второй производной, в частности, в точке максимума вторая производная строго меньше нуля.
В данном случае для функции V(l) имеем:
Первая производная обращается в ноль при l = b и l = b/3 - это возможные точки экстремума. Для второй производной имеем V"(b) = 2ab > 0 и V"(b/3) = -2ab < 0. Следовательно, l = b - точка минимума, а l = b/3 - точка максимума.
Итак, при l = b/3 объём балки будет наибольшим и равным 4ab[sup]3[/sup]/27.
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