To determine the average rate of change (or speed) of the distance runner during the interval from 0.75 hours to 1.00 hours, we need to follow a step-by-step process:
1. Identify the relevant points:
- Time at the start of the interval ([tex]\( t_1 \)[/tex]): 0.75 hours.
- Distance at the start of the interval ([tex]\( d_1 \)[/tex]): 3.50 miles.
- Time at the end of the interval ([tex]\( t_2 \)[/tex]): 1.00 hours.
- Distance at the end of the interval ([tex]\( d_2 \)[/tex]): 4.75 miles.
2. Calculate the change in time ([tex]\( \Delta t \)[/tex]):
[tex]\[
\Delta t = t_2 - t_1 = 1.00 \, \text{hours} - 0.75 \, \text{hours} = 0.25 \, \text{hours}
\][/tex]
3. Calculate the change in distance ([tex]\( \Delta d \)[/tex]):
[tex]\[
\Delta d = d_2 - d_1 = 4.75 \, \text{miles} - 3.50 \, \text{miles} = 1.25 \, \text{miles}
\][/tex]
4. Determine the average rate of change (speed):
[tex]\[
\text{Average rate of change} = \frac{\Delta d}{\Delta t} = \frac{1.25 \, \text{miles}}{0.25 \, \text{hours}}
\][/tex]
5. Simplify the result:
[tex]\[
\frac{1.25 \, \text{miles}}{0.25 \, \text{hours}} = 5.0 \, \text{miles per hour}
\][/tex]
Therefore, the average rate of change, or the runner's speed, during the interval from 0.75 to 1.00 hours is [tex]\(5.0\)[/tex] miles per hour.
Thus, the correct answer is [tex]\( \boxed{5.00 \, \text{miles per hour}} \)[/tex].
