the table shows the profit from a school book fair based on the number of books sold.\nprofit vs. books…

Answer:

A. $0.50 per book

Explanation:

Step1: Recall rate - of - change formula

The rate of change (slope) of a function $y = f(x)$ is given by $m=\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$.

Step2: Choose two points

Let $(x_1,f(x_1))=(100,50)$ and $(x_2,f(x_2))=(250,275)$.

Step3: Calculate the rate of change

$m=\frac{275 - 50}{250-100}=\frac{225}{150}=1.5$. Let's check with another pair. Let $(x_1,f(x_1))=(250,275)$ and $(x_2,f(x_2))=(300,350)$. Then $m=\frac{350 - 275}{300 - 250}=\frac{75}{50}=1.5$. Let's use the first and the last points: $(x_1,f(x_1))=(100,50)$ and $(x_2,f(x_2))=(350,425)$. Then $m=\frac{425 - 50}{350 - 100}=\frac{375}{250}=1.5$. But if we consider the change in profit per book, we can also note that when the number of books sold increases by 50 (e.g., from 300 to 350), the profit increases by $425 - 350=75$. So the rate of change $\frac{75}{50}=1.5$ dollars per book. However, if we assume there is a fixed - cost component and we calculate the marginal profit (rate of change of profit with respect to number of books sold), we can use the formula. Let's take two points $(x_1,f(x_1))=(100,50)$ and $(x_2,f(x_2))=(250,275)$. The rate of change $m=\frac{275 - 50}{250 - 100}=\frac{225}{150}=1.5$. If we consider the linear relationship $y=mx + b$ and calculate the change in $y$ over the change in $x$ for any two points from the table. For example, if we take $(x_1,f(x_1))=(100,50)$ and $(x_2,f(x_2))=(300,350)$: [m=\frac{350 - 50}{300 - 100}=\frac{300}{200}=1.5] The rate of change of the function (profit per book) is $$1.50$ per book.