Question

1. find the rate of change from the graph. explain what the rate of change means in each situation.

a. distance sound travels in air b. speed

Explanation:

Step1: Calculate the rate of change for part a

The formula for the rate of change (slope) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Let \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(70,14)\).
\(m=\frac{14 - 0}{70 - 0}=\frac{14}{70}=\frac{1}{5}\) miles per second.
This means that for every second that passes, the sound travels \(\frac{1}{5}\) of a mile.

Step2: Calculate the rate of change for part b

Let \((x_1,y_1)=(0,128)\) and \((x_2,y_2)=(7,0)\).
\(m=\frac{0 - 128}{7 - 0}=-\frac{128}{7}\approx - 18.29\) ft/s per second.
The negative rate of change means that the speed is decreasing. For every second that passes, the speed decreases by approximately \(18.29\) feet per second.

Answer:

a. The rate of change is \(\frac{1}{5}\) miles per second. It means the sound travels \(\frac{1}{5}\) mile each second.
b. The rate of change is \(-\frac{128}{7}\approx - 18.29\) ft/s per second. It means the speed decreases by approximately \(18.29\) ft/s each second.