Question

On Friday, Kaitlyn went to the park to work out. She started with a slow jog at a constant rate for five minutes. She then increased her speed to a faster, constant pace for seven minutes. She took a three-minute break, then ran for another 7.5 minutes at a constant pace to finish her two-mile run. She rested for the remainder of the time. (2 points) During what domain interval will she be at 1.25 miles?

Answer 1

To determine the time interval during which Kaitlyn will be at 1.25 miles, let's break down her workout step by step based on the information given.

1. First Phase: Slow Jog (5 minutes)

   • Assume she jogs at a slow pace. We'll denote her slow jog speed as \( v_1 \) miles per minute.
   • We don’t know \( v_1 \) yet, but let's call the distance she covers in this phase \( d_1 \). In 5 minutes, she covers: \[ d_1 = v_1 \times 5 \]

2. Second Phase: Faster Pace (7 minutes)

   • Assume she increases her speed to a faster pace, \( v_2 \) miles per minute.
   • In this case, the distance she covers in this second phase is \( d_2 \): \[ d_2 = v_2 \times 7 \]

3. Third Phase: Break (3 minutes)

   • She takes a break here, which doesn’t contribute to any distance covered.

4. Fourth Phase: Final Run (7.5 minutes)

   • After the break, she runs for 7.5 minutes at a constant pace, which we'll denote as \( v_3 \). The distance she covers in this phase is: \[ d_3 = v_3 \times 7.5 \]

The total distance she runs is \( d_1 + d_2 + d_3 = 2 \) miles.

To find the interval during which she is at 1.25 miles, we should figure out when she reaches this distance.

Working Through Phases

1. Through Slow Jog (5 minutes):

   • In this phase, terminates at \( \text{Distance}_1 = d_1 \).

2. Through Faster Pace (7 minutes):

   • After 5 + 7 = 12 minutes, the total distance becomes \( d_1 + d_2 \).

3. Through Break (3 minutes):

   • Total time now is 15 minutes, and the total distance still remains \( d_1 + d_2 \).

4. Through Final Run (7.5 minutes):

   • After the break, at 15 + 7.5 = 22.5 minutes, she finishes with a total distance of 2 miles.

We need to find:

Distance Calculations

Assuming she reaches the halfway point (1 mile) within the first two intervals before taking a break, we can deduce:

At the end of 5 minutes:

• Let's say she jogged at a pace of 0.5 miles/min, covering: \[ d_1 = 0.5 \times 5 = 2.5 \text{ miles (not possible as the maximum is 2 miles).} \] Let’s analyze her progress:

If Kaitlyn covered 1 mile after the two activities before her break, the portion of the distance when she runs can be:

• Let's assume her jog was relatively moderate and she could run a total of 1 mile by around 15 (5 + 7 minutes).

• If she runs fast enough to hit this by 10-12 minutes distance would approximate \( \frac{1.0}{15} \text{ miles per minute.} \) Once we equalize 1.25 mile intervals:

• Before the run: She would complete 1 mile at around 12 minutes. She should maintain a constant pace.

• Assuming constant speed: 1.25 mile in corresponding intervals; running a fast pace this may happen around minute 20 to 22.5.

Conclusion

Interval for 1.25 miles will be approximately between: 15 minutes and 22.5 minutes post the 3 minutes stretch after run finishing at 22.5 minutes. Thus, she should be at 1.25 miles between approximately 15 to 20 minutes.