Find the dimensions that minimize cost if the box has a volume of 45 cubic feet.

Question

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $10 per square foot. Find the dimensions that minimize cost if the box has a volume of 45 cubic feet.

Length of base 

 Height of side  

Answer 1

Let x = length of a side to the square base of the box.

Let y = length of the height of the box.

The Volume of the box = x*x*y = (x²)*(y) = yx². We are told that the volume is $45. Hence, the two variables x and y, are related to each other as follows:

yx² = 45

The cost of the material used for the top and bottom is $5 per ft². The area of the top is x², so the cost of the top is 5x². The bottom costs the same: 5x² because it is made of the same material and has the same dimensions.

The cost of the material use for the sides is $10 per square ft². The area of each side is x*y, and there are four sides. Thus the cost of the sides is 10 * x*y * 4 = 40xy.

The total cost of the material used to construct the box is C(x,y) = 5x² + 5x² + 40xy = 10x² + 40xy.

We seek to determine the values of x and y that minimize C(x,y). To do that, we first solve for y:

y = 45 / x²

We now substitute for y in our cost function:

C(x,y) = 10x² + 40xy

C(x) = 10x² + 40x*(45/x²)

= 10x² + 1800/x

It is instructive to look at a graph of C(x) vs. x:

https://www.desmos.com/calculator/4cs2qadtvb

From the graph, we see the minimum cost is achieved at x = 4.481 ft.

From y = 45 / x², we get that the minimum cost is achieved when y = 45 / 4.481² = 2.2411 ft.

Having arrived at the graphical answer, let us determine the answer using Calculus:

C(x) = 10x² + 1800/x

C'(x) = 20x - 1800/x² = 0

or

20x = 1800/x²

Simplifying, we get

x³ = 90

x = 4.481, same as before.

Answer 2

If you setup the box on the x-y-z axis' and the top & bottom are square, then x = y

The volume = 45 = x²z or z = 45/x²

The cost function will then be 10x² + 40xz or 10x² + 1800/x

Taking the derivative of this one variable function will give 20x - 1800/x² = 0 or

x = 4.4814 = y

z = 2.2407