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]
