Question 630·Medium·Math

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The monthly profit, $P$ , in thousands of dollars, that a digital magazine earns from subscriptions can be modeled by

P(n) = -0.2n^2 + 3.2n - 7,

where $n$ is the subscription fee, in dollars, charged to each customer. According to the model, what subscription fee maximizes the magazine’s monthly profit?

A quadratic profit model with a negative coefficient on the squared fee makes a downward-opening graph. Its turning point is the highest point, called the vertex. Graph the model in Desmos and click that point. Read the fee from the horizontal coordinate; the vertical coordinate is the profit, which answers a different question.

Hints

Hint 1
The input goes on the horizontal axis, while the output goes on the vertical axis. Here the subscription fee is the input. Which coordinate of the highest point gives you that fee?
Hint 2
A negative coefficient on the squared input makes a parabola open downward. Its vertex, where the curve turns, is its highest point. Graph the model to locate that point.

Step-by-step

Graph the profit model
Step 1
Identify what the question asks for
The fee $n$ is the input, and $P(n)$ is the monthly profit. So you want the fee where $P(n)$ is greatest, not the greatest profit amount.
Step 2
Locate the maximum on the graph
The coefficient of $n^2$ is negative ( $-0.2$ ), so the graph opens downward. Its vertex, the point where the curve turns, is its highest point.
Step 3
Read the fee at the vertex
Type $P(x)=-0.2x^2+3.2x-7$ in Desmos, using $x$ for the fee $n$ . Click the curve, then its gray point at the top. Desmos shows $(8,5.8)$ . The first coordinate is the fee; the second is profit in thousands of dollars. To find what maximizes profit, read the input at the peak, not the height of the peak. The subscription fee is $8. Choice B.

Question 630·Medium·Math

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