For each demand equation, express the total revenue R as a function of the price p p...

For each demand equation, express the total revenue R as a function of the price p per item, sketch the graph of the resulting function, and determine the price p that maximizes total revenue in each case. HINT [See Example 3.]

Example 3:

Alien Publications, Inc. predicts that the demand equation for the sale of its latest illustrated sci-fi novel Episode 93: Yoda vs. Alien is

q = −2,000p + 150,000

where q is the number of books it can sell each year at a price of $p per book. What price should Alien Publications, Inc., charge to obtain the maximum annual revenue?

Solution The total revenue depends on the price, as follows:

This value of p gives the highest point on the graph and thus gives the largest value of R(p). We may conclude that Alien Publications, Inc., should charge $37.50 per book to maximize its annual revenue.

q = −3p + 300