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.

Question

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.