Question

PLEASE HELP!!!!

If 300 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas. For example, if the answer is 2,000 write 2000.
Please help and explain how to do it.

Answer 1

500

Explanation:

It's a box with a square base, so let's say the width and length are x and the height is y.

The surface area of the box without the top is:

A = x² + 4xy

300 = x² + 4xy

The volume of the box is:

V = x²y

Solve for y in the first equation and substitute into the second:

y = (300 − x²) / 4x

V = x² (300 − x²) / 4x

V = x (300 − x²) / 4

V = 75x − ¼ x³

To optimize V, find dV/dx and set to 0:

dV/dx = 75 − ¾ x²

0 = 75 − ¾ x²

x = 10

So the volume of the box is:

V = 75x − ¼ x³

V = 500

The maximum volume is 500 cm³.