Applied Max and Min

Question

If 600 cm² of material is available to make a box with a square base and a closed 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.

Answer 1

Let 600cm² = 2x² +4xh and solve for h to get h=(600- 2x²)/4x. Use this in the volume formula V=x²h to get V= x²(600-2x²)/4x. This gives V= 150x- x³/2 and dV/dx = 150 -3x²/2 =0 and x = 10 cm and using this in the expression for h gives h=10 cm. So the maximum volume is 1000 cm³.