Question 871·Medium·Math

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A concert promoter sets the ticket price at $p$ dollars. The number of tickets that can be sold at that price is modeled by $n = 800 - 10p$ , where $0 \le p \le 80$ . The revenue, in dollars, from ticket sales is given by the function

R(p) = p(800 - 10p), \qquad R(p) > 0.

Which of the following is the best interpretation of the factor $800 - 10p$ in this context?

In a revenue model, revenue is price per ticket times tickets sold. If a factor also appears in an equation that defines another quantity, use that equation to identify the factor. Then check the units: dollars per ticket times tickets gives dollars. Don’t mistake one factor for the whole revenue.

Hints

Hint 1
An equivalent expression has the same value as the expression it replaces. Look for an expression that appears in both given equations. How could substituting it connect the two models?
Hint 2
A unit check tests an interpretation by comparing the units of a product with those of its factors. How could you use the units of revenue and ticket price to test the remaining factor’s meaning?

Step-by-step

Match the factor to its defined quantity
Step 1
Rewrite revenue using $n$
No Desmos needed. The question asks what a factor means, not for a calculation.

The model says $n=800-10p$ . Replace that matching factor in $R(p)=p(800-10p)$ with $n$ :
$R(p)=pn.$
The substitution works because $n$ and $800-10p$ name the same quantity.
Step 2
Identify what each factor measures
Here $p$ is dollars per ticket, and $n$ counts tickets. Their product is the revenue in dollars, so the factor equal to $n$ , $800-10p$ , is the number of tickets sold. Read a factor from its defining equation, not from the meaning of the whole product. Choice D.

Question 871·Medium·Math

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