Question

the graph shows the amount of money miguel earns after working x hours.
amount earned vs. hours worked
what is the rate of change of the amount earned with respect to hours worked for this function?
options:

• $\frac{1}{13}$ hours per dollar
• $\frac{2}{5}$ hours per dollar
• 3 dollars per hour
• 13 dollars per hour

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 \(\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(y\) is amount earned (dollars), \(x\) is hours worked. We use points \((2, 26)\) and \((5, 65)\).

Step2: Calculate the slope

Substitute into the formula: \(\frac{65 - 26}{5 - 2}=\frac{39}{3} = 13\)? Wait, no, wait. Wait, rate of change of amount earned (y) with respect to hours (x) is \(\frac{\Delta y}{\Delta x}\). Wait, \(y\) is dollars, \(x\) is hours. So slope is \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{65 - 26}{5 - 2}=\frac{39}{3}=13\)? Wait, no, wait, let's check the points. Wait, (2,26): 2 hours, 26 dollars. (5,65): 5 hours, 65 dollars. So change in y (dollars) is 65 - 26 = 39, change in x (hours) is 5 - 2 = 3. So rate of change of amount earned (dollars) with respect to hours is \(\frac{39}{3}=13\) dollars per hour? Wait, no, wait: slope is \(\frac{\Delta y}{\Delta x}\), where \(y\) is amount earned (dollars), \(x\) is hours. So \(\frac{65 - 26}{5 - 2}=\frac{39}{3}=13\) dollars per hour. Wait, but let's check with (0,0) and (2,26): \(\frac{26 - 0}{2 - 0}=13\). Yes, so the rate of change is 13 dollars per hour. Wait, but let's check the options. Option D is 13 dollars per hour. Wait, but wait, maybe I mixed up. Wait, no: rate of change of amount earned (y) with respect to hours (x) is \(\frac{y}{x}\) for linear function. At x=2, y=26: 26/2=13. At x=5, y=65: 65/5=13. So yes, slope is 13 dollars per hour.

Answer:

13 dollars per hour (the option: 13 dollars per hour)