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?

Question

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

Answer 1

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 2

Thats not right

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!