Spencer G. answered • 10/13/21

Mechanical Engineering Graduate Student

To first solve this problem we need to setup two equations to quantify both volume and price.

Volume = 20 = x^2 * y

Price = x^2 * (0.19) + 4x * y * (0.06) + x^2 * (0.11)

Price = 0.3 * x^2 + 0.24 * x * y

Now with these equations we can find the derivative of price to determine the best selling region. But we must substitute volume into the equation as well.

x^2 * y = 20

y = 20/x^2

Price = 0.3*x^2 + 0.24*x*(20/x^2)

Price = 0.3*x^2 + 4.8/x

Taking the derivative and setting to zero:

P' = 0.6*x - 4.8/x^2

0 = 0.6*x - 4.8/x^2

x^3 = 8

x = 2

We now know that the slope goes from negative to positive at x = 2 which means it is the lowest value of price for the box. Now solving for y using the volume equation we get:

x = 2 feet

y = 5 feet