Revenue Pack-Em-In Real Estate is building a new housing development. The...

Revenue Pack-Em-In Real Estate is building a new housing development. The more houses it builds, the less people will be willing to pay, due to the crowding and smaller lot sizes. In fact, if it builds 40 houses in this particular development, it can sell them for $200,000 each, but if it builds 60 houses, it will only be able to get $160,000 each. Obtain a linear demand equation and hence determine how many houses Pack- Em-In should build to get the largest revenue. What is the largest possible revenue? 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:

We are after the price p that gives the maximum possible revenue. Notice that what we have is a quadratic function of the form where a = −2,000, b = 150,000, and c = 0. Because a is negative, the graph of the function is a parabola, concave down, so its vertex is its highest point (Figure 5). The p coordinate of the vertex is

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.