Find the dimensions of the box that minimize cost.

Question

You are constructing a box for your cat to sleep in. The plush material for the square bottom of the box costs $5/ft² and the material for the sides costs $2/ft². You need a box with volume 4 ft². Find the dimensions of the box that minimize cost.

Any help would be amazing thanks

Answer 1

The formula for the volume of a rectangular prism is

V= l•w•h

Since the bottom is square, therefore l=w. We can rewrite the equation:

V= w²•h

The volume is 4 cubic feet:

4 = w²•h

h = 4/w²

The cost of the bottom = 5w²

The cost of the 4 rectangle sides = 2•4wh = 8wh

The total cost (C) will be:

C = 5w² + 8wh

Substitute the value of h in terms of w:

C = 5w² + 8w(4/w²)

C = 5w² + 32/w

Get the derivative of C with respect to w:

dC/dw = 10w - 32/w²

To minimize, dC/dw = 0

10w - 32/w² = 0

(10w³ -32)/w² = 0

10w³ -32 = 0

10w³ = 32

w³ = 32/10 = 16/5 = 3.2

w = (3.2)^1/3 ≈ 1.47

h = 4/(3.2)^2/3 ≈ 1.84

The length and width should be both approximately equal to 1.47 ft. and the height should be approximately equal to 1.84 ft.