The image shows a problem involving estimating distance traveled by a runner over time by using speed data at specific intervals. The table provided gives the runner's speed at 0, 15, 30, 45, 60, and 75 minutes, and the problem requires calculating underestimates and overestimates of distance using rectangles.

To solve these:

1. Underestimate for the First Half Hour (0–30 minutes):

   • Divide the 30-minute interval into two segments (0–15 minutes and 15–30 minutes).
   • Use the lower speed for each interval to get an underestimate:
     • 0–15 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
     • 15–30 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
   • Total Underestimate = $3.25 + 3.25 = 6.5 \text{ miles}$

2. Overestimate for the First Half Hour (0–30 minutes):

   • Use the higher speed in each interval:
     • 0–15 min: $15 \text{ min} \times 14 \text{ mph} = 3.5 \text{ miles}$
     • 15–30 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
   • Total Overestimate = $3.5 + 3.25 = 6.75 \text{ miles}$

3. Underestimate for the Whole Time (0–75 minutes):

   • Calculate each 15-minute segment using the lower speed:
     • 0–15 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
     • 15–30 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
     • 30–45 min: $15 \text{ min} \times 8 \text{ mph} = 2 \text{ miles}$
     • 45–60 min: $15 \text{ min} \times 4 \text{ mph} = 1 \text{ mile}$
     • 60–75 min: $15 \text{ min} \times 0 \text{ mph} = 0 \text{ miles}$
   • Total Underestimate = $3.25 + 3.25 + 2 + 1 + 0 = 9.5 \text{ miles}$

4. Overestimate for the Whole Time (0–75 minutes):

   • Use the highest speed for each interval:
     • 0–15 min: $15 \text{ min} \times 14 \text{ mph} = 3.5 \text{ miles}$
     • 15–30 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
     • 30–45 min: $15 \text{ min} \times 13 \text{ mph} = 3.25 \text{ miles}$
     • 45–60 min: $15 \text{ min} \times 8 \text{ mph} = 2 \text{ miles}$
     • 60–75 min: $15 \text{ min} \times 4 \text{ mph} = 1 \text{ mile}$
   • Total Overestimate = $3.5 + 3.25 + 3.25 + 2 + 1 = 13 \text{ miles}$

Would you like more details on each calculation step or an explanation of how to interpret rectangles in these estimations?

Here are 5 related questions to delve deeper:

1. How would the estimates change if the intervals were shorter?
2. Can you calculate the exact distance if the speed curve is known?
3. How would you interpret the difference between overestimates and underestimates?
4. What happens to the estimates if the runner’s speed increases towards the end?
5. How could integral calculus provide a more precise answer?

Tip: When using rectangles for distance estimation, the choice of height (speed) in each interval determines whether it's an overestimate or underestimate.