Question

A box will be built with a square base and an open top. Material for the base costs $8 per square foot, while material for the sides cost $2 per square foot. Find the dimensions of the box of maximum volume that can be built for $2400. I'll give a medal. Help please

Answer 1

give some names to the dimensions of the box.
It will have a square base, so let x be the length of the sides of this square
let y be the height.

write down a formula for the volume of the box:
V= x*x*y
write down a formula for the cost of the box:
cost of the base is 8x^2
cost of one side is 2*x*y. there are 4 sides. so 8xy for the 4 sides
There is no top.

Cost= 8x^2 +8xy = 2400
solve for y:
\[ y = \frac{2400-8x^2}{8x} \]
use that value for y in the equation for volume
once you have x, use one of the above equations to find y

Now take the derivative, set equal to zero, and solve for x

Answer 2

Thank you!!