Optimization

Question

An open-top shipping crate with square bottom and rectangular sides is to hold 256 ft^3 and requires a minimum amount of cardboard. Find the most economical dimensions. Hint: your cardboard is for the 4 sides and the bottom only. 

Answer 1

Since the base is a square, we'll represent each side as x. We'll let the height of the box be y. Using the volume formula we get the relation x²y=256. Solving this for y we get y=256/x², which we will substitute in later.

 

The material needed to make the box is x² for the bottom and 4xy for the 4 sides. So the total material needed is

SA=x²+4xy.

 

You can now replace y to get a function of x which would be SA=x²+4x(256/x²) or SA=x²+1024/x or SA=x²+1024x^-1.

Now take the derivative:

SA' = 2x -1024x^-2     Set it equal to 0 and solve 

2x - 1024/x² = 0

2x³ - 1024 = 0

x³ - 512 = 0

x³ = 512

x = 8

 

Plug back in to find y......y=256/64 = 4.

 

Hope this helped.