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. During what domain interval will she reach 1.25 miles?

Question

(1 point)
Responses

15≤s≤22.5
15 less-than-or-equal-to s less-than-or-equal-to 22.5

12≤s≤15
12 less-than-or-equal-to s less-than-or-equal-to 15

22.5≤s≤25
22.5 less-than-or-equal-to s less-than-or-equal-to 25

5≤s<12

Answer 1

To determine during what interval Kaitlyn will reach 1.25 miles during her workout, we need to analyze her activity step by step:

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

   • Let's denote her slow jog speed as \( s_1 \).
   • Since we don't have her exact speed yet, we can't calculate the distance she covers in this segment. However, this is the first segment.

2. Second stage (Faster pace for 7 minutes):

   • Let's denote her faster jog speed as \( s_2 \).
   • Again, we don’t know this speed yet.

3. Break (3 minutes):

   • No distance is covered during this break.

4. Final stage (Run for 7.5 minutes):

   • Let's denote her final run speed as \( s_3 \).

Now, if she covered a total of 2 miles, we need to calculate at what point she reaches 1.25 miles.

Approach:

1. Calculate the distance covered in each stage:

   • For the first stage (5 minutes): Distance = \( s_1 \times \frac{5}{60} \) miles
   • For the second stage (7 minutes): Distance = \( s_2 \times \frac{7}{60} \) miles
   • For the third stage (7.5 minutes): Distance = \( s_3 \times \frac{7.5}{60} \) miles

2. Total Distance = 2 miles: The total can be written as: \[ \left(s_1 \times \frac{5}{60}\right) + \left(s_2 \times \frac{7}{60}\right) + \left(s_3 \times \frac{7.5}{60}\right) = 2 \]

3. We should determine when she reaches 1.25 miles. Since she will reach 1.25 miles at some point during any of the stages before she covers the total of 2 miles, we can set up the following:

   • During the slow jog: \[ \text{Distance covered in 5 minutes} = s_1 \times \frac{5}{60} \]

   • During the faster pace (adding to previous distance): \[ \text{Distance covered in 12 minutes total} = s_1 \times \frac{5}{60} + s_2 \times \frac{7}{60} \]

   • During the run post-break (adding to previous total distance): \[ \text{Distance covered in 19.5 minutes} = \left[s_1 \times \frac{5}{60} + s_2 \times \frac{7}{60}\right] + s_3 \times \frac{7.5}{60} \]

Maximizing estimates and considering typical paces, Kaitlyn would reach 1.25 miles sometime between her slow and fast run stages. The critical part is understanding where in time those stages would cover that distance.

Given the provided intervals:

• 15 ≤ s ≤ 22.5 (would relate to more advanced running speeds)
• 12 ≤ s ≤ 15 (likely represents moderate to fast jogging)
• 22.5 ≤ s ≤ 25 (High-performance sprint appearance)
• 5 ≤ s < 12 (Very slow pace – likely not reached 1.25 mile)

Assuming average jog speeds: Given 2 miles in approximately 25.5 minutes suggests she's often exceeding 4.7 mph. Most of her run stages would indeed suggest she'd be reaching 1.25 assertively within the range of 15 ≤ s ≤ 22.5.

Final Answer: 15 ≤ s ≤ 22.5