Question

A company is designing shipping crates and wants the volume of each crate to be 4 cubic feet, and the crate's base to be a square between 1 and 1.5 feet per side. If the material for the bottom costs $5, the sides $3 and the top $1 per square foot. What dimensions will give the minimum cost?

Answer 1

______
/_____/| $5 bottom
|| || $1 Top
|| || H $3 Side
||____||
|/____|/S
S

A = S^2 * H Cost will be related
4 = S^2 * H with Surface Area
H = 4 / S^2 So our equation is...

SA = 2S^2 + 4SH
Cost = 5S^2 + S^2 + 4*3*S(4/S^2)
Cost = 6S^2 + 48/S

Now we differentiate Cost

Cost' = 12S - 48/S^2

Now we must determine where the derivative of cost is equal to 0

0 = 12S - 48/S^2
48/S^2 = 12S
48 = 12S^3
S = 4^(1/3)

So a critical number will occur at the cube root of 4. Do a number line analysis.

Domain: [1,1.5]
-- -- ++
|----------|-----------|-------
1 1.5 4^(1/3)

The local minimum occurs at 4^(1/3), which is not in the domain situation. Instead, use 1.5 for the value of S.

H = 4/S^2
H = 4/[3/2]^2
so H = 16/9

Answer: A 1.5 x 1.5 x 16/9 box will minimize the cost.