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?
Christian traveled 1.5 miles from \square \leq t \leq \square

Answer 1

Thats not right

Answer 2

To determine the time interval during which Christian traveled 1.5 miles, we first need to analyze his running schedule and speed.

1. **Jogging for 5 minutes**: Let's denote his jogging speed as \( v_1 \) (in miles per minute). The distance he jogs in this time is:
\[
d_1 = v_1 \times 5
\]

2. **Running at a faster pace for 7 minutes**: We denote his faster speed as \( v_2 \) (in miles per minute). The distance during this segment is:
\[
d_2 = v_2 \times 7
\]

3. **Total distance traveled when he takes a break**: After these two segments, the total distance covered is:
\[
d_{\text{total}} = d_1 + d_2 = v_1 \times 5 + v_2 \times 7
\]

4. **Resting for 4 minutes**: During this time, no distance is covered.

Next, we will assume some reasonable values for his speeds since they are not given:

- Traditional jogging speed \( v_1 \) might be around 0.1 miles/minute (6 mph).
- Faster running speed \( v_2 \) could be around 0.2 miles/minute (12 mph).

Calculating distances:

1. **Jogging**:
\[
d_1 = 0.1 \text{ miles/min} \times 5 \text{ min} = 0.5 \text{ miles}
\]

2. **Running**:
\[
d_2 = 0.2 \text{ miles/min} \times 7 \text{ min} = 1.4 \text{ miles}
\]

Adding the two distances gives:
\[
d_{\text{total}} = 0.5 + 1.4 = 1.9 \text{ miles}
\]

**Identifying when he covered 1.5 miles**:
- During the first 5 minutes (jogging), he only covered 0.5 miles, so he has not reached 1.5 miles yet.
- In the next 7 minutes, he continued running, thus covering 1.4 more miles:

Adding the 0.5 miles jogged to the distance covered in the 7-minute run:
\[
\text{By 5 min: } 0.5 \text{ miles}
\]
\[
\text{By 12 min (5 min + 7 min): } 0.5 + 1.4 = 1.9 \text{ miles}
\]
Since we need to find the point at which he reaches exactly 1.5 miles, we will calculate the exact timing during the faster run.

He needs to cover:
\[
1.5 - 0.5 = 1 \text{ mile}
\]

At his faster pace of 0.2 miles/minute:
\[
\text{Time to reach 1 mile} = \frac{1 \text{ mile}}{0.2 \text{ miles/min}} = 5 \text{ minutes}
\]

Since he started running at the 5th minute and continued for 7 minutes, he reaches 1.5 miles at:
\[
\text{Starting point at 5 min} + 5 min = 10 min.
\]

Thus, the interval in which Christian traveled 1.5 miles is:
\[
5 \leq t \leq 10.
\]

So, the final answer is:
**Christian traveled 1.5 miles from \( 5 \leq t \leq 10 \)**.

Answer 3

I apologize for any confusion in my previous explanation. Let's analyze the problem step-by-step to correctly determine the time interval when Christian traveled 1.5 miles.

### Breakdown of the Run:

1. **Jogging Phase:**
- Duration: 5 minutes.
- Let's assume Christian jogs at a speed of \( v_1 \) miles per minute.

2. **Faster Pace Phase:**
- Duration: 7 minutes.
- Let's assume during this phase, he runs at a speed of \( v_2 \) miles per minute.

3. **Rest Phase:**
- Duration: 4 minutes (no distance covered).

### Steps to Calculate Distance:

1. **Distance during Jogging:**
\[
d_1 = v_1 \cdot 5 \text{ minutes}
\]

2. **Distance during Faster Running:**
\[
d_2 = v_2 \cdot 7 \text{ minutes}
\]

### Total Time Duration:
The total time of the run (including rest) is \( 5 + 7 + 4 = 16 \) minutes.

### Goal:
We want to find when Christian has covered a total distance of **1.5 miles**.

### Determine Time Intervals:

1. **At the end of the Jogging Phase (5 minutes):**
- Total distance covered: \( d_1 = v_1 \cdot 5 \)
- Assume a realistic jogging speed, for example, \( v_1 = 0.1 \) miles/minute.
- Distance after 5 minutes = \( 0.1 \text{ miles/min} \cdot 5 \text{ min} = 0.5 \text{ miles} \).

2. **During the Faster Pace Phase (5 minutes to 12 minutes):**
- Assume a higher running speed, for example, \( v_2 = 0.2 \) miles/minute.
- Distance during the first 5 minutes of this phase (from 5 to 10 minutes):
\[
d_2 = v_2 \cdot 5 = 0.2 \text{ miles/min} \cdot 5 \text{ min} = 1 \text{ mile}
\]
- Total distance covered by 10 minutes:
\[
0.5 \text{ miles} + 1 \text{ mile} = 1.5 \text{ miles}
\]

3. **Conclusion:**
- He reaches **1.5 miles exactly at 10 minutes.**
- He continues to run from 10 to 12 minutes, covering additional distance.

### Final Answer:
Christian traveled 1.5 miles during the time interval:
\[
5 \leq t \leq 10.
\]

If there is a specific speed example you'd like to provide to clarify or adjust any assumptions, please let me know!