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

5≤s<12
5
≤
𝑠
<
12
5 less-than-or-equal-to s less than 12

15≤s≤22.5

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

12≤s≤15

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

22.5≤s≤25

Answer 1

To determine the interval during which Kaitlyn reaches 1.25 miles, let's break down her workout into segments and calculate her distances and times.

1. Slow jog for 5 minutes:
   Let \( r_1 \) be her speed during the slow jog. The distance covered is: \[ \text{Distance}_1 = r_1 \times 5 \text{ (in minutes)} \] We convert minutes into hours (5 minutes = \( \frac{5}{60} = \frac{1}{12} \) hours): \[ \text{Distance}_1 = r_1 \times \frac{1}{12} \]

2. Faster pace for 7 minutes:
   Let \( r_2 \) be her speed during the faster pace. The distance is: \[ \text{Distance}_2 = r_2 \times 7 \text{ (in minutes)} \] Convert 7 minutes into hours (7 minutes = \( \frac{7}{60} \) hours): \[ \text{Distance}_2 = r_2 \times \frac{7}{60} \]

3. Break for 3 minutes:
   No distance covered.

4. Final run for 7.5 minutes:
   Let \( r_3 \) be her speed during this segment. The distance is: \[ \text{Distance}_3 = r_3 \times 7.5 \text{ (in minutes)} \] Convert 7.5 minutes into hours (7.5 minutes = \( \frac{7.5}{60} = \frac{1.25}{10} = \frac{1}{8} \) hours): \[ \text{Distance}_3 = r_3 \times \frac{1}{8} \]

Now we know her total distance is 2 miles, so: \[ \text{Distance}_1 + \text{Distance}_2 + \text{Distance}_3 = 2 \] Given this equation setup, we will calculate how far she goes in each segment until she reaches 1.25 miles.

Timeline:

1. During the slow jog (0 to 5 minutes):
   She runs \( \text{Distance}_1 = r_1 \times \frac{1}{12} \). Let’s assume \( r_1 = v \), then: \[ \text{Distance}_1 = v \times \frac{1}{12} \]

2. During the faster pace (5 to 12 minutes):
   She runs for another 7 minutes, moving towards 1.25 miles.

The first 5 minutes + 7 minutes = 12 minutes total.

Let’s say she runs at a constant speed:

Using the simplifications for clarity:

• After the first 5 minutes: \( v \times \frac{1}{12} \)
• After the next 7 minutes, \( v_2 \), where \( v_1, v_2 \) leads us to reach or exceed 1.25 miles.

If we calculate to approximate: Using the average, assume she reaches: 1 mile in the 12 minutes (let's assume a moderate pace) gives ( 0.75 miles, reaching 1.25 mile during the slower or faster period, most probably in 7.5 minutes as it reaches the full capacity to get to 2 miles.

Concluding

Thus when reaching under a 15-min segment (beyond the threshold of her jog) she moves into the domain interval of her speed setting at time frames:

Based upon the answers:

• Therefore, she reaches 1.25 miles at about 12 ≤ s < 15. The most appropriate response is:

12≤s≤15.