Question

the table shows the profit from a school book fair based on the number of books sold. profit vs. books sold books sold (x) profit f(x) 100 $50.00 250 $275.00 300 $350.00 350 $425.00 what is the rate of change for the function represented in the table? $0.50 per book $0.67 per book $1.07 per book $1.50 per book

Explanation:

Step1: Recall rate of change formula

The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). Let's take the first two points \((100, 50)\) and \((250, 275)\).

Step2: Calculate the difference in profit and books sold

\(y_2 - y_1 = 275 - 50 = 225\), \(x_2 - x_1 = 250 - 100 = 150\).

Step3: Compute the rate of change

Rate of change \(=\frac{225}{150}= 1.5\)? Wait, no, wait. Wait, let's check another pair. Take \((250, 275)\) and \((300, 350)\). \(y_2 - y_1 = 350 - 275 = 75\), \(x_2 - x_1 = 300 - 250 = 50\). \(\frac{75}{50}=1.5\). Another pair: \((300, 350)\) and \((350, 425)\). \(425 - 350 = 75\), \(350 - 300 = 50\). \(\frac{75}{50}=1.5\). Wait, but first pair: \((100,50)\) and \((250,275)\): \(\frac{275 - 50}{250 - 100}=\frac{225}{150}=1.5\). Oh, right. So the rate of change is \(1.50\) per book. Wait, but let's check the first point: if \(x = 100\), \(f(x)=50\). If rate is \(1.5\) per book, then for \(x = 100\), \(y = 1.5\times100 + b\). \(50 = 150 + b\), so \(b=-100\). Then for \(x = 250\), \(y = 1.5\times250 - 100 = 375 - 100 = 275\), which matches. For \(x = 300\), \(1.5\times300 - 100 = 450 - 100 = 350\), matches. For \(x = 350\), \(1.5\times350 - 100 = 525 - 100 = 425\), matches. So the rate of change is \(1.50\) per book. Wait, but the options have \$1.50 per book. Wait, but let's check the first calculation again. Wait, maybe I made a mistake earlier. Wait, the first pair: (100,50) and (250,275). The change in y is 275 - 50 = 225, change in x is 250 - 100 = 150. 225 divided by 150 is 1.5. Yes. So the rate of change is 1.50 per book.

Answer:

\$1.50 per book (the option: \$1.50 per book)