Question

Kristen went on a run he jogged for 5 minutes then increased his rate to a faster constant pace for 7 minutes he then took a rest for 4 minutes during what time interval has Kristen traveled 1.5 MI
Kristen traveled 1.5 miles from __ < to < __

Answer 1

To determine the time interval during which Kristen traveled 1.5 miles, we need to break down the components of his run.

1. **Jogging for 5 minutes**:
- We need the speed for this part to know how far he traveled.
- Let's denote his jogging speed as \( S_1 \) (in miles per minute).

2. **Faster constant pace for 7 minutes**:
- We'll denote this speed as \( S_2 \) (in miles per minute), which is faster than \( S_1 \).

3. **Rest for 4 minutes**:
- During this time, he does not travel any distance.

Let's assume some hypothetical speeds for illustration, since the exact speeds aren’t provided:

- If we assume Kristen jogs at a speed of 5 minutes per mile (12 mph) during the first segment, then his distance for those 5 minutes is:
\[
\text{Distance} = \text{Speed} \times \text{Time} = \frac{1 \text{ mile}}{5 \text{ minutes}} \times 5 \text{ minutes} = 1 \text{ mile}
\]

- If during the faster pace (7 minutes), he runs at 3 minutes per mile (20 mph), then:
\[
\text{Distance} = \frac{1 \text{ mile}}{3 \text{ minutes}} \times 7 \text{ minutes} \approx 2.33 \text{ miles}
\]

In this example, he travels 1 mile during the jog and then covers another 2.33 miles during the faster run. He reaches the 1.5-mile mark sometime during these segments.

To find the exact time interval:

1. **In the first 5 minutes**, he hasn’t reached 1.5 miles yet (only 1 mile).
2. **In the next 7 minutes**, he reaches:
- At the end of 5 minutes: 1 mile.
- After the first minute of the fast pace (6 minutes total): \(1 + \frac{1}{3} = 1.33\) miles.
- After another 6 minutes (up to 12 minutes): \(1 + 2.33 = 3.33\) miles.

Now, how far does he need to run from the 5-minute mark to reach 1.5 miles?

He needs an additional \(0.5\) miles after reaching 1 mile. At the faster pace (3 minutes per mile), it will take:
\[
\text{Time} = \text{Distance} \times \text{Rate} = 0.5 \times 3 = 1.5 \text{ minutes}
\]
So, he will reach 1.5 miles at:
\[
5 \text{ minutes} + 1.5 \text{ minutes} = 6.5 \text{ minutes} \text{ during the faster pace}.
\]

Thus, Kristen travels 1.5 miles from:
\[
5 < \text{time} < 6.5 \text{ minutes}.
\]

In conclusion, to express Kristen's travel interval in other words, he traveled 1.5 miles during the time interval of **5 minutes to 6.5 minutes**.