SubsectionRevenue

In this section we consider some applications that lead to quadratic equations. The first application involves revenue.

Revenue is the amount of money a company takes in from selling a product. To find the total revenue from the sale of a product, we multiply the price of one item by the number of items sold.

Revenue $=$ (price per item) $\cdot$ (number of items sold)

For example, if a snack bar sells 30 sandwiches at $4.25 each, their revenue from sandwiches is \begin{equation*} \text{Revenue} = ($4.25~ \text{per sandwich}) \cdot (30~ \text{sandwiches}) = $127.50 \end{equation*} Look Ahead. Usually, the number of items that consumers will buy depends on the price of the item: the higher a company sets the price, the fewer items it is likely to sell. How does a company's revenue depend upon the price it charges? A good way to study such a problem is to consider the graph of the equation for revenue. Example6.8. Rick works as a personal trainer at his gym. By experimenting with his fee, he discovers that if he charges $p$ dollars per hour, he attracts $40-p$ clients. 1. Write a quadratic equation for Rick's revenue, $R\text{,}$ in terms of his fee, $p\text{.}$ 2. Graph your equation for $R\text{.}$ 3. What should Rick charge if he wants to make$300 revenue? How many clients will he attract?
4. Use your graph to find the fee Rick should charge in order to earn the largest possible revenue. How many clients will he attract at that fee?
Solution
1. Rick's revenue is given by

\begin{align*} \text{Revenue} \amp = \text{(hourly fee)} \cdot \text{(number of clients)}\\ R \amp = p(40-p) = 40p - p^2 \end{align*}
2. We make a table of values and plot the points to obtain the parabola shown in the figure.

 $p$ $R$ $\hphantom{00}$ $10$ $300$ $40(\alert{10})-\alert{10}^2$ $20$ $400$ $40(\alert{20})-\alert{20}^2$ $30$ $300$ $40(\alert{30})-\alert{30}^2$ $40$ $0$ $40(\alert{40})-\alert{40}^2$

28.

The height in feet of a football $t$ seconds after being kicked from the ground is given by

\begin{equation*} h=-16t^2+80t \end{equation*}
1. Factor the expression for $h\text{.}$
2. Complete the table and sketch a graph.

 $t$ $h$ $0$ $\hphantom{0000}$ $1$ $\hphantom{0000}$ $2$ $\hphantom{0000}$ $2.5$ $\hphantom{0000}$ $3$ $\hphantom{0000}$ $4$ $\hphantom{0000}$ $5$ $\hphantom{0000}$

3. What is the maximum height of the football? When does it reach this height?
4. When does the football fall back to the ground?
29.

Sportsworld sells $180-3p$ pairs of their name-brand running shoes per week when they charge $p$ dollars per pair.

1. Write an equation for Sportsworld's revenue, $R\text{,}$ in terms of $p\text{.}$
2. Fill in the table and graph the equation.

 $p$ $R$ $0$ $\hphantom{0000}$ $10$ $\hphantom{0000}$ $20$ $\hphantom{0000}$ $30$ $\hphantom{0000}$ $40$ $\hphantom{0000}$ $50$ $\hphantom{0000}$ $60$ $\hphantom{0000}$
3. At what price(s) will Sportsworld's revenue be zero?
4. At what price will Sportsworld's revenue be maximum? What is their maximum revenue?

For Problems 30–32, find the $x$-intercepts of the graph of the equation.

30.

$y=(3x-4)(x+2)$

31.

$y=3x^2+12x$

32.

$y=-7x-4x^2$

For Problems 33–34, graph all three equations on the same grid. What do you observe?

33.

$y=x$

$y=6-x$

$y=6x-x^2$

34.

$y=x+3$

$y=3-x$

$y=x^2-9$

For Problems 35–36, $\alert{\text{find the mistake}}$ in the steps for solution, then write a correct solution.

35.

\begin{aligned}[t] 4x^2 \amp = 12x\\ \dfrac{4x^2}{4x} \amp = \dfrac{12x}{4x}\\ x \amp = 3e \end{aligned}

36.

\begin{aligned}[t] 9x^2 - 4 \amp = 16\\ 3x-2 \amp = 4\\ 3x \amp = 6\\ x \amp = 2\\ \end{aligned}