To determine the time interval during which Christian traveled 1.5 miles, we first need to analyze his running schedule and speed.
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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: \[ d_1 = v_1 \times 5 \]
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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: \[ d_2 = v_2 \times 7 \]
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Total distance traveled when he takes a break: After these two segments, the total distance covered is: \[ d_{\text{total}} = d_1 + d_2 = v_1 \times 5 + v_2 \times 7 \]
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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:
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Jogging: \[ d_1 = 0.1 \text{ miles/min} \times 5 \text{ min} = 0.5 \text{ miles} \]
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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 \( 5 \leq t \leq 10 \).