Suppose a box with a square base and open top has a volume of 32 cubic feet. Calculus?
Suppose a box with a square base and open top has a volume of 32 cubic feet. Find the dimensions of the box that minimize the amount of the material used.
Thank you so much in advance. Studying for finals and having trouble with the steps of this problem 
Let the length of the square base be x ft
Let the height be y ft
So we have volume of the box x^2 * y = 32 cubic feet
This will help us get a relation between x and y.
For the material used, we have to consider the five sides of the box (no top, assuming that the floor and sides are equally thick)
So material = x^2 + 4*xy
putting it just in terms of x, material = x^2 + 4*32/x
To minimize, differentiate Material w.r.t. x and equate to 0:
2x – (4*32)/x^2 = 0
2x = 128/x^2
x^3 = 64
or x = 4.
Since x = 4, y =2.
Material used = 4*4+ 4*(4*2)
= 80 square feet
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