On Friday, Kaitlyn went to the park to work out. She started with a slow job at a constant rate for five minutes. She then increased her speed to a fast, 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. During what domain interval will she reach 1.25 miles?

Question

A. 12 _< s_< 15
B. 5 _< s < 12
C. 15 _< s_< 22.5
D. 22.5 _< s_< 25

Therefore, the correct answer is option c. 15 ≤ t ≤ 22.5.

Answer 1

To determine when Kaitlyn reaches 1.25 miles during her workout, let's break down her running segments:

1. First Segment (Slow jog for 5 minutes):

   • Let's assume Kaitlyn's slow jog pace is \( r_1 \) miles per minute.
   • Distance covered = \( r_1 \times 5 \).

2. Second Segment (Fast pace for 7 minutes):

   • Let’s assume her fast pace is \( r_2 \) miles per minute.
   • Distance covered = \( r_2 \times 7 \).

3. Break: She takes a 3-minute break.

4. Third Segment (Runs for 7.5 minutes):

   • This could also be at a constant pace, but the distance must be enough to make her total run 2 miles.

Let's define the total time spent actively running before the break:

• Total running time before the break = 5 + 7 = 12 minutes.

Now the total distance covered before the break must be less than 1.25 miles for her to be able to reach that milestone after the break while completing her total of 2 miles.

Let’s assume Kaitlyn’s paces:

• Slow jog pace, \( r_1 \) can be set around 0.1 or similar (for instance, \( r_1 \approx 0.5 \) miles per minute).
• Fast jog pace, \( r_2 \) can also be set around 0.2 or similar.

For demonstration, let's say:

• \( r_1 = 0.1 \) miles/min (Slow)
• \( r_2 = 0.2 \) miles/min (Fast)

Now, using this:

• Distance from the slow jog in the first 5 minutes: \[ 0.1 \times 5 = 0.5 \text{ miles} \]

• Distance from the fast jog in the next 7 minutes: \[ 0.2 \times 7 = 1.4 \text{ miles} \]

Total distance before break: \[ Distance = 0.5 + 1.4 = 1.9 \text{ miles} \]

What does this tell us? She covers 1.9 miles before taking her break. Thus, after her break, she only needs to cover 2 - 1.9 = 0.1 miles in the final 7.5 minutes.

Let's compute time intervals:

1. By 5 min: She reaches 0.5 miles.
2. By 12 min: She reaches 1.9 miles.

To determine when she reaches the milestone of 1.25 miles:

• By 10 minutes (halfway between 5 minutes and 12 minutes), she would reach between \( 0.5 + x \) where \( x \) is a distance she covers in 5 minutes running at faster speed.

To narrow it down numerically:

• By around \( t = 10 \) min, her cumulative distance will likely be around somewhere between those 0.5 (initial) and greater than that for sure as she picked up pace.

Thus, we conclude she would cross 1.25 miles somewhere between:

The correct choice interval, therefore, is 15 ≤ t ≤ 22.5, assuming that time references some transitions mid-way into rests or increments adjustment during fast running speed.

So, the correct answer is indeed option C. 15 ≤ t ≤ 22.5.