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

Answer 2

So the hard part here is at the start of the problem, and actually requires a skill you're taught more in Algebra than in Calculus. And that's the task of setting up a system of equations.

In this case, given we let x represent the length and width of the box and y represent the height, we have two equations:

Price:

P = 5x^2 + 4*2xy

(The 4 in front of the 2xy refers to the fact that there are four sides; there is no 2 in front of 5x^2 because it's an open box (we want the cat to live, right?).)

Volume (l*w*h):

V = x^2*y

or

4 = x^2*y, because we're told the volume is 4.

Remember, though--we're only minimizing price. And it would be useful to have that in one variable, so that's what we'll do.

First, using my volume equation to solve for y:

4/x^2 = y.

Now, substituting y for x in my price equation:

P = 5x^2 + 4*2x*4/x^2

Simplifying a bit...

P = 5x^2 + 32/x

Remember, though, we're looking for a max and min. Places where the first derivative equals zero, in other words. So, using our power rule:

P' = 10x - 32/x^2

Then, using what we know about optimization (specifically, that a max or min occurs where the first derivative equals zero), we set the derivative equal to zero, then solve:

0 = 10x - 32/x^2

0 = (10x^3 - 32)/x^2 (by getting a common denominator)

0 = 10x^3 - 32

32 = 10x^3

32/10 = x^3

And by taking the cube root, we get:

1.47 (rounded) = x

But one last, easy part remains: finding y (which we let represent the height of the box).

We'll do this by substituting x back in to our volume equation:

4 = x^2*y

4 = 1.47^2*y

1.84 = y.

To answer the question, then: this box has a base that is approximately 1.47 feet long, 1.47 feet wide, and is 1.84 feet tall.

I hope that helped!