A theater models its weekly revenue R, in dollars, from... | SAT Nonlinear equations in one variable and systems of equations in two variables Question, Answer Explained

Question bank / Nonlinear equations in one variable and systems of equations in two variables

Math · Advanced MathMedium

A theater models its weekly revenue $R$ , in dollars, from ticket sales as $R = p(120 - 2p)$ , where $p$ is the price of a ticket, in dollars. Two different ticket prices give a weekly revenue of $\$1{,}600$ . What is the lower of these two prices, in dollars?

$20$

$30$

$40$

$60$

Show answer and explanation

Answer: A

Set $p(120 - 2p) = 1{,}600$ : $120p - 2p^2 = 1{,}600$ . Rearrange and divide by $-2$ : $p^2 - 60p + 800 = 0$ . Factor: $(p - 20)(p - 40) = 0$ , so $p = 20$ or $p = 40$ . The lower price is $\$20$ .

Choice B, $30$ , is the price that gives the maximum revenue, $\$1{,}800$ , not a price that gives $\$1{,}600$ .

Why the other choices are wrong

B. A price of $30 gives the maximum revenue, 30 × 60 = $1,800, not $1,600.
C. $40 also gives $1,600 in revenue, but it is the higher of the two prices.
D. At a price of $60, the revenue is 60 × 0 = $0.

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