Question

Боковые стороны и меньшее основание трапеции имеют одинаковые длины – по 50 см. Найдите размер ее большего основания, при котором площадь трапеции была бы наибольшей. Тема - применение производной для отыскания наибольших и наименьших значений величин

Answer 1

 Большее основание положим равно $x$  так как трапеция равнобедренная , то  высота 
 $H=\sqrt{50^2-(\frac{x-50}{2})^2}\\\\ S=\frac{x+50}{2}*\sqrt{50^2-(\frac{x-50}{2})^2}\\\\ S=\frac{x+50}{2}*\frac{\sqrt{2500*4-(x-50)^2}}{2}$
 Рассмотрим функцию   
 $S=\frac{x+50}{2}*\frac{\sqrt{2500*4-(x-50)^2}}{2}\\ S=\frac{x+50}{4}*\sqrt{7500-x^2+100x}\\\\ S'=\frac{ 10000-(x-50)^2-x^2+2500}{4\sqrt{10000-(x-50)^2}}\\\\ S'=0\\\\ 10000-2x^2+100x=0\\\\ x=100\\\\ x=-50$
   Функция  убывает на отрезке  $x \in [100;150)$ 
 откуда  следует что большее основание должно равняться  $x=100$ см

Answer 2

Finding the Maximum Area of a Trapezoid

To find the size of the larger base of a trapezoid that maximizes its area, we can use the concept of derivatives. The area of a trapezoid is given by the formula:

Area = (1/2) * (sum of the bases) * height

In this case, we know that the lengths of the lateral sides and the smaller base of the trapezoid are all 50 cm. Let's denote the larger base as 'x' cm. The height of the trapezoid is not given, so we'll keep it as 'h' cm.

Using the given information, we can set up the equation for the area of the trapezoid:

Area = (1/2) * (50 + x) * h

To find the value of 'x' that maximizes the area, we need to find the derivative of the area function with respect to 'x' and set it equal to zero. Let's differentiate the area function:

d(Area)/dx = (1/2) * h * (d(50 + x)/dx)

Since 'h' is a constant, we can ignore it for now. Differentiating '50 + x' with respect to 'x' gives us:

d(50 + x)/dx = 1

Therefore, the derivative of the area function simplifies to:

d(Area)/dx = (1/2)

Setting this derivative equal to zero and solving for 'x', we find that there is no critical point. This means that the area of the trapezoid is always increasing, and there is no maximum value for the area.

In conclusion, the area of the trapezoid will continue to increase as the larger base 'x' increases. There is no specific value of 'x' that maximizes the area.

Please let me know if you need any further clarification or assistance!