BUSINESS CALCULUS

Question

Find two nonnegative numbers whose sum is 16 and so that the product of one number and the square of the other number is a maximum. Round the answer to three significant digits. 

Answer 1

This is a standard optimization problem. Assuming your two numbers are x and y, are trying to ensure that the product of x and y^2 is a maximum. In other words, P = xy^2 is the equation you're maximizing.

Now because the two numbers have to add up to 16, you can express that relationship as x+y=16. With this in mind, you can express x in terms of y as x = 16-y.

Plugging that expression back into P = xy^2 for x, you'll get the single-variable function P = (16-y)y^2 or P = 16y^2-y^3

Because you are looking for the values that yield the maximum product, you'll want to look at the critical points of the new P equation. To do that find the derivative. By the power rule, you should get the following:

P' = 32y - 3y^2

or

P' = y(32-3y)

By setting this equation to zero, you should get y = 0 and y = 32/3.

Because x = 16-y, you have the corresponding values of x = 16 and x = 16-32/3 = 16/3. To determine which of the two pairs yields the maximum, simply plug them into the original expression P = xy^2. The larger of the two products will be the maximum.