Question

the table shows how far a distance runner has traveled since the race began. what is her average rate of change, in miles per hour, during the interval 0.75 to 1.00 hours?

time elapsed (hours)	miles traveled (miles)
0.75	3.50
1.00	4.75

options: 5.00 miles per hour, 4.75 miles per hour, 6.00 miles per hour, 5.50 miles per hour

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). In this case, \( x \) is time (in hours) and \( y \) is miles traveled. So we need to find the change in miles traveled divided by the change in time between \( t = 0.75 \) hours and \( t=1.00 \) hours.

Step2: Identify the values at \( t = 0.75 \) and \( t = 1.00 \)

From the table, when \( t = 0.75 \) hours, the miles traveled \( f(0.75)=3.50 \) miles. When \( t = 1.00 \) hours, the miles traveled \( f(1.00) = 4.75 \) miles. The time interval is \( b - a=1.00 - 0.75 = 0.25 \) hours.

Step3: Calculate the change in miles and the average rate of change

First, calculate the change in miles: \( f(1.00)-f(0.75)=4.75 - 3.50=1.25 \) miles. Then, use the average rate of change formula: \(\frac{1.25}{0.25}\).

Step4: Simplify the fraction

\(\frac{1.25}{0.25}=\frac{125}{25}=5\). Wait, that's not right? Wait, no, wait: \( 4.75 - 3.50 = 1.25 \), and \( 1.00 - 0.75=0.25 \). Then \( 1.25\div0.25 = 5 \)? Wait, but let's check again. Wait, maybe I made a mistake. Wait, \( 4.75 - 3.50 = 1.25 \), \( 1 - 0.75 = 0.25 \). \( 1.25/0.25 = 5 \)? Wait, but the options have 5.00, 4.75, 6.00, 5.50. Wait, maybe I misread the table. Wait, the table: at 0.75 hours, 3.50 miles; at 1.00 hours, 4.75 miles. So the change in distance is \( 4.75 - 3.50 = 1.25 \) miles. Change in time is \( 1.00 - 0.75 = 0.25 \) hours. So average rate of change is \( \frac{1.25}{0.25}=5 \) miles per hour. Wait, but let me check again. Wait, \( 1.25\div0.25 \): 0.25 times 5 is 1.25. Yes. So the average rate of change is 5.00 miles per hour.

Wait, but maybe I made a mistake. Wait, let's recalculate:

\( f(b)-f(a)=4.75 - 3.50 = 1.25 \)

\( b - a=1.00 - 0.75 = 0.25 \)

Average rate of change \(=\frac{1.25}{0.25}=5\)

Yes, that's correct. So the average rate of change is 5.00 miles per hour.

Answer:

5.00 miles per hour