Question

Christian went on a run. He jogged for five minutes, then increased his rate to a faster constant pace for seven minutes. He then took a rest for four minutes. During what time interval has Christian traveled 1.5 miles?

(1 point)
Christian traveled 1.5 miles from
≤t≤
.

Answer 1

To find the time interval during which Christian traveled 1.5 miles, we first need to determine his pace during jogging and his faster constant pace.

1. **Assumptions about pace**:
- Let's assume Christian jogged at a pace of 6 miles per hour during the first 5 minutes.
- For the second segment, let's assume his faster pace was 9 miles per hour for 7 minutes.

2. **Calculating distance**:
- The distance covered while jogging for 5 minutes:
\[
\text{Distance} = \text{Speed} \times \text{Time} = 6 \, \text{miles/hour} \times \left(\frac{5}{60} \, \text{hours}\right) = 0.5 \, \text{miles}
\]

- The distance covered at the faster pace for 7 minutes:
\[
\text{Distance} = 9 \, \text{miles/hour} \times \left(\frac{7}{60} \, \text{hours}\right) = 1.05 \, \text{miles}
\]

3. **Finding total distance**:
- By the end of the jog and the faster pace:
\[
\text{Total Distance} = 0.5 \, \text{miles} + 1.05 \, \text{miles} = 1.55 \, \text{miles}
\]

4. **Determining the time intervals**:
- **From 0 to 5 minutes**: Christian travels 0.5 miles.
- **From 5 to 12 minutes (5 + 7)**: He travels an additional 1.05 miles, totaling 1.55 miles by the end of 12 minutes.

5. **Finding the time interval for 1.5 miles**:
- He covers 1.5 miles between:
- At 5 minutes he has traveled 0.5 miles
- As he continues running, we can set up a proportion.

The additional distance he needs to reach 1.5 miles is:
\[
1.5 \, \text{miles} - 0.5 \, \text{miles} = 1 \, \text{mile}
\]

At his faster pace, it takes him:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1 \, \text{mile}}{9 \, \text{miles/hour}} = \frac{1}{9} \, \text{hours} \approx 6.67 \, \text{minutes}
\]

6. **Calculating the interval**:
- Therefore, he travels 1.5 miles at approximately:
\[
t \approx 5 \, \text{minutes} + 6.67 \, \text{minutes} = 11.67 \, \text{minutes} \approx 11 \, \text{minutes} \, 40 \, \text{seconds}
\]

So, to summarize:

Christian traveled 1.5 miles from \(0 \leq t \leq 11.67\) minutes.

Final answer:
Christian traveled 1.5 miles from \(5 \, \text{minutes} \leq t \leq 11.67 \, \text{minutes}\).