karinalo23
Answered

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?

\begin{tabular}{|c|c|}
\hline
Time Elapsed (Hours) & Miles Traveled (Miles) \\
\hline
0.50 & 2.00 \\
\hline
0.75 & 3.50 \\
\hline
1.00 & 4.75 \\
\hline
\end{tabular}

A. 4.75 miles per hour
B. 5.00 miles per hour
C. 5.50 miles per hour
D. 6.00 miles per hour



Answer :

GinnyAnswer
To find the average rate of change in miles per hour during the interval from 0.75 to 1.00 hours for the distance runner, follow these steps:

1. Identify the initial and final times within the given interval:
[tex]\[ \text{Initial Time ( \( t_{\text{initial}} \) )} = 0.75 \text{ hours} \][/tex]
[tex]\[ \text{Final Time ( \( t_{\text{final}} \) )} = 1.00 \text{ hours} \][/tex]

2. Identify the distance (miles) traveled at these corresponding times:
[tex]\[ \text{Miles at \( t_{\text{initial}} \) } = 3.50 \text{ miles} \][/tex]
[tex]\[ \text{Miles at \( t_{\text{final}} \) } = 4.75 \text{ miles} \][/tex]

3. Calculate the change in distance ([tex]\( \Delta \text{Miles} \)[/tex]):
[tex]\[ \Delta \text{Miles} = \text{Miles at \( t_{\text{final}} \)} - \text{Miles at \( t_{\text{initial}} \)} \][/tex]
[tex]\[ \Delta \text{Miles} = 4.75 \text{ miles} - 3.50 \text{ miles} = 1.25 \text{ miles} \][/tex]

4. Calculate the change in time ([tex]\( \Delta \text{Time} \)[/tex]):
[tex]\[ \Delta \text{Time} = t_{\text{final}} - t_{\text{initial}} \][/tex]
[tex]\[ \Delta \text{Time} = 1.00 \text{ hours} - 0.75 \text{ hours} = 0.25 \text{ hours} \][/tex]

5. Determine the average rate of change (miles per hour) by dividing the change in distance by the change in time:
[tex]\[ \text{Average Rate of Change} = \frac{\Delta \text{Miles}}{\Delta \text{Time}} \][/tex]
[tex]\[ \frac{1.25 \text{ miles}}{0.25 \text{ hours}} = 5.0 \text{ miles per hour} \][/tex]

So, the average rate of change for the distance runner during the interval from 0.75 to 1.00 hours is:
[tex]\(\boxed{5.0 \text{ miles per hour}}\)[/tex]

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