Volume and minimum cost

Question

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.37 per square foot, the material for the sides costs $0.05 per square foot, and the material for the top costs $0.13 per square foot, determine the dimensions of the box that can be constructed at minimum cost.
Length=?, width=?, height=?

Answer 1

Let x represent the measure of a side of the square base.

The volume of a rectangular box is given by V = LWH.

Substitute into that formula where L and W are both x.

40 ft³ = (x)(x)(h) where h is the height of the box.

Now solve for h in terms of x.

h = 40/x²

Let's set up the cost function using pennies/cents (instead of fractions of a dollar).

The area of the base is x² and each square foot costs 37 cents: 37x² represents the cost of the base in terms of x.

Similarly for the cost of the top @ 13 cents per square foot: 13x².

The cost of the top and bottom combined: 50x².

Each side of the box is a rectangle with a width of x and a height of h where h = 40/x².

Each side has an area (40/x²) (x) = 40/x = 40x^-1.

Each side is 5 cents per square foot: 200x^-1.

There are 4 sides: 800x^-1 represents the cost of the 4 sides in total.

The cost function is:

C(x) = 50x² + 800x^-1 {x > 0}

To determine the value of x that generates a minimum cost:

C'(x) = 100x - 800x^-2

C''(x) = 100 +1600x^-3

When does the 1st derivative equal zero?

100x = 800/x²

100x³ = 800

x³ = 8

x = 2

C''(2) = some positive number, so C is concave up at x = 2, meaning x = 2 generates a minimum value for C.

When x = 2, h = 40/4 = 10.

The cost (minimum) when x = 2 is 600 cents or $6.

The dimensions of the box that generate a minimum cost:

A square base with side length 2 and a height 10 (feet).

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