Question

Use the revenue function, R(x) = -9x² + 198x, and cost function, C(x) = 90x + 180, where x is the number of items made

and sold, to determine each of the following. Assume both revenue and cost are given in dollars. Enter multiple answers
using a comma-separated list when necessary.

Find the number of items sold when revenue is maximized.

Answer 1

To find the number of items sold that maximizes revenue, the vertex formula x = -b/(2a) is used on the revenue function R(x) = -9x² + 198x, resulting in 11 items as the quantity that maximizes revenue.

Step-by-step explanation:

We are given a revenue function R(x) = -9x² + 198x and a cost function C(x) = 90x + 180, where x represents the number of items made and sold.

To maximize revenue, we must determine the value of x that gives the highest point on the revenue function, which is equivalent to finding the vertex of the parabolic revenue function.

We know that a parabola in the form ax² + bx + c has its vertex at x = -b/(2a).

Applying this to our revenue function, we calculate the number of items sold when revenue is maximized using:

x = -198 / (2 * (-9)) = 11.

Therefore, when 11 items are sold, the revenue is maximized.