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 _≤ t ≤_

Answer 1

To calculate the time interval during which Christian traveled 1.5 miles, we need to determine the rates at which he jogged and ran.

Let's define:
- \( r_1 \): Christian's jogging pace (in miles per minute).
- \( r_2 \): Christian's faster running pace (in miles per minute).

**Step 1: Jogging Phase**
- He jogs for 5 minutes.
- Distance traveled while jogging:
\[
d_1 = r_1 \times 5
\]

**Step 2: Running Phase**
- He runs for 7 minutes.
- Distance traveled while running:
\[
d_2 = r_2 \times 7
\]

**Step 3: Combined Distance**
The total distance traveled when he finishes running (before resting) is:
\[
d_{\text{total}} = d_1 + d_2 = r_1 \times 5 + r_2 \times 7
\]

**Step 4: Rest Phase**
During the 4-minute rest period, he does not cover any distance; thus, the distance remains the same.

**Step 5: Determining the Time Interval**
To find the time interval in which he traveled 1.5 miles, we need to consider the scenarios in which 1.5 miles was reached. This will depend on the values of \( r_1 \) and \( r_2 \).

Suppose Christian jogged at a rate of \( r_1 \) = 0.5 miles/minute (a common jogging speed). In that case:
\[
d_1 = 0.5 \times 5 = 2.5 \, \text{miles}
\]
It's clear he already exceeded 1.5 miles before the end of his jog.

Thus, for the purposes of our interval, assume he jogged until he reached 1.5 miles.

**Calculating Interval for 1.5 miles:**
1. If he jogs for \( t \) minutes at \( r_1 \):
\[
\text{During jogging: } d = r_1 \times t \leq 1.5
\]
For \( r_1 = 0.5 \):
\[
0.5t = 1.5 \implies t = 3 \, \text{minutes}
\]

Since he jogs for 5 minutes, he reaches 1.5 miles at some point in the first 3 minutes.

2. He will continue running and not exceed 1.5 miles until 5 minutes when the jog ends; thus, the 1.5-mile criterion will be reached here.

After reaching his distance, he changes his steps to a faster run and will continue to add distance.

Taking into consideration the overall motion:
- Beginning of the jog is at \( t = 0 \)
- \( 3 \leq t \leq 5 \) while jogging, where he covers 1.5 miles before the end of jog.

So, based on the entire scenario and observations, the final answer would read:

Christian traveled 1.5 miles from \( 0 \leq t \leq 5 \).
As he reaches and surpasses the distance of 1.5 miles during this entire initial jog phase.