To determine the time interval in which Christian traveled 1.5 miles, we need to break down what we know about his running and rest periods.
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Jogging for 5 minutes: Let's assume Christian jogs at a specific pace. If we denote his jogging pace as \( r_1 \) miles per minute, then in 5 minutes he will have traveled: \[ d_1 = r_1 \times 5 \]
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Running at a faster pace for 7 minutes: Let’s denote his faster running pace as \( r_2 \) miles per minute. During this time, he travels: \[ d_2 = r_2 \times 7 \]
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Resting for 4 minutes: During this period, he does not travel any distance.
Now, we want to find the time interval where he has traveled a total of 1.5 miles. He jogged and then ran, and we will add distances for those times.
Total Distance Calculation: The total distance \( d \) that Christian has traveled after his run would be: \[ d = d_1 + d_2 = (r_1 \times 5) + (r_2 \times 7) \]
To determine when he first reaches 1.5 miles, we might want to analyze both portions of his run:
Assuming \( r_1 \) and \( r_2 \) are such that in the 5 minutes of jogging (from time \( t = 0 \) to \( t = 5 \)), he covers a part of the distance, and then in the 7 minutes of running (from \( t = 5 \) to \( t = 12 \)), he covers additional distance.
If he hasn't reached 1.5 miles by the end of his jog, we can consider the following:
- At \( t = 5 \) min, the distance is \( r_1 \times 5 \).
- His distance at the end of the run (12 min total) will be: \[ d = (r_1 \times 5) + (r_2 \times 7) \]
Let's assess a scenario:
Example:
Consider the following hypothetical speeds:
- \( r_1 = 0.1 \) miles/minute (jogging speed)
- \( r_2 = 0.25 \) miles/minute (running speed)
Distance Calculation:
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Jogging (first 5 minutes): \[ d_1 = 0.1 \times 5 = 0.5 \text{ miles} \]
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Running (next 7 minutes): \[ d_2 = 0.25 \times 7 = 1.75 \text{ miles} \]
Now we add both: \[ d = d_1 + d_2 = 0.5 + 1.75 = 2.25 \text{ miles} \] This means:
- At \( t = 5 \) he reached 0.5 miles.
- At \( t = 12 \), he reached 2.25 miles.
To find when he reaches 1.5 miles, notice that after he jogged for 5 minutes (0.5 miles): In the next 7 minutes, he runs at a speed of 0.25 miles/minute:
- By 6 minutes, he has traveled \( 0.5 + 0.25 \times 1 = 0.75 \) miles.
- By 7 minutes, \( 0.5 + 0.25 \times 2 = 1.0 \) miles.
- By 8 minutes, \( 0.5 + 0.25 \times 3 = 1.25 \) miles.
- By 9 minutes, \( 0.5 + 0.25 \times 4 = 1.5 \) miles.
Time Interval:
Thus, Christian reaches 1.5 miles sometime between:
- Starting at the 8th minute (1.25 miles)
- Ending at the 9th minute (1.5 miles)
Conclusively: \[ \text{Christian traveled 1.5 miles from } 8 \leq t \leq 9 \text{ minutes}. \]