Answer:
H=28.3 cm
Explanation:
The shape of a vertical trough and its cross-section are shown in the images below. Note the width of the rectangular sheet of metal (80 cm) is folded into two equal sides of 40 cm each which form the hypotenuse of the triangle where the dimensions of the shape will be determined.
As indicated in the question, the volume of the trough is its area times the length:

The cross-section is a triangle whose area can be computed as

The sides H and X are related by using the Pythagora's formula

Solving for X

The area is now

And the volume is

Since L=1000

To find the maximum volume, we compute the first derivative
![V'=500(H'√(1600-H^2)+H[√(1600-H^2)]'](https://img.qammunity.org/2021/formulas/mathematics/college/w9askvg400hlz4b4khjhyrf6g0z9j1v3gu.png)

Equating V'=0

Operating

Solving for H
