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
For me option C is the correct. But I need to check with you. Thanks

Answer 1

To determine when Kaitlyn reaches 1.25 miles, we first need to break down her running segments and calculate how far she runs in each segment.

1. First Segment: Kaitlyn jogged for 5 minutes at a slow pace. Let's denote her slow pace as \( s_1 \) (in minutes per mile). We don't know the exact speed, but we will use it to find her distance later.

2. Second Segment: She then ran at a fast pace for 7 minutes. We'll denote her fast pace as \( s_2 \) (in minutes per mile).

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

4. Third Segment: She then ran for 7.5 minutes at another constant pace until she reached the 2 miles total.

Let's use the information we have to calculate her distances.

Assuming Kaitlyn maintains a slow pace of 12 minutes per mile for \( s_1 \) during the first segment:

• Distance covered in the first 5 minutes: \[ \text{Distance}_1 = \frac{5}{12} \text{ miles} \approx 0.4167 \text{ miles} \]

Now, she runs at a faster pace (we'll assume 8 minutes per mile for \( s_2 \)) for 7 minutes:

• Distance covered in the second segment: \[ \text{Distance}_2 = \frac{7}{8} \text{ miles} = 0.875 \text{ miles} \]

Total distance before the break: \[ \text{Total Distance before break} = \text{Distance}_1 + \text{Distance}_2 \approx 0.4167 + 0.875 \approx 1.2917 \text{ miles} \]

Now, she takes a 3-minute break, which means no distance is covered during this time. After the break, she runs for 7.5 minutes. The pace for the final run must be fast enough to complete a total of 2 miles.

Total distance she still needs to cover: \[ \text{Distance to finish} = 2 - 1.2917 \approx 0.7083 \text{ miles} \]

If she runs the last segment in 7.5 minutes, her pace must be calculated for this distance: \[ \text{Required pace} = \frac{7.5}{0.7083} \approx 10.59 \text{ minutes per mile} \]

Now calculating when she reaches 1.25 miles:

At 1.25 miles:

• Up to 0.4167 miles (5 mins)
• She needs \( 1.25 - 0.4167 = 0.8333 \) miles more after that jogging distance.

She runs 0.875 miles (in 7 minutes): \[ \text{0.875 miles is enough to reach 1.25 miles, we must calculate exact minute for 1.25}
\]

To find how long it takes her to reach 1.25 miles in that last running segment:

• \( \text{Needed distance to reach 1.25 miles} = 1.25 - 0.4167 \), so she has to run 0.8333 miles.

Using her speed in the 7 minutes (ideal if she maintained that performance):

• Distance covered to that point must be during her fast run, meaning we need to check how fast she was going.
• \( \text{Total duration to reach} = 5 + 7 + \text{time needed to run 0.4167} + 3 = \text{duration} \).

Since the distance of 0.833 can be determined with a rough timescale given her initial intervals before break shows reached within the above minutes. The exact position of hitting 1.25 is most likely taking 12mins to 15 under metric speed. That leads to:

Final Answer: The correct domain interval for when Kaitlyn reaches 1.25 miles is A. \( 12 < s < 15 \).

Thus option A is potentially the correct choice instead of C.

Answer 2

In order to find my domain interval:
- when she will reach 1.25 miles, we need to determine the time it takes her to reach that distance.
The distance covered during the slow jog is given by: d = r1 × t1
Since she runs at a constant rate, we can use the formula d = r × t. So, we can write the equation for the slow jog as: 1.25 = r1 × 5
Solving for r1, we get r1 = 1.25/5

= 0.25 miles per minute.

After this, I calculated the total time for the slow jog and the faster pace: 5 + 7 = 12 minutes.
The distance covered during the faster pace is given by: d = r2 × t2

Using formula d = r × t, we can write the equation for the faster pace as: 1.25 - 0.25 = r2 × 7
Solving for r2, we get r2 = 1/7

= 0.1429 miles per minute.
Since Kaitlyn takes a 3-minute break, we subtract this time from the total time to determine the time it takes to reach 1.25 miles: 12 - 3 = 9 minutes.

Finally, we calculate the distance covered during the last part of the run: d = r3 × t3

Using the formula d = r × t, we can write the equation for the last part of the run as: 1.25 - 1 = r3 × 7.5

Solving for r3, we get r3 = 0.1667 miles per minute.

Now, we can determine the domain interval when Kaitlyn will reach 1.25 miles. From the calculations above, we know that it will take her 9 minutes to reach that distance.

So my option:. 15 ≤ s ≤ 22.5.

Answer 3

Let's clarify the situation and calculations for when Kaitlyn reaches 1.25 miles. The steps you've provided describe each phase of her workout. Let's break it down more systematically to see how we reach the point at which she covers 1.25 miles.

Breakdown of Kaitlyn's Workout:

1. First Segment (Slow Jog):

   • Time = 5 minutes
   • Let's assume her slow jog pace is \( r_1 \) (miles per minute).
   • Distance covered in this segment: \[ d_1 = r_1 \times 5 \]

2. Second Segment (Fast Pace):

   • Time = 7 minutes
   • Let's assume her fast pace is \( r_2 \) (miles per minute).
   • Distance covered in this segment: \[ d_2 = r_2 \times 7 \]

3. Break:

   • Time = 3 minutes (no distance covered).

4. Third Segment (Final Run):

   • Time = 7.5 minutes
   • Let's assume her pace for this segment is \( r_3 \).
   • Distance covered in this segment: \[ d_3 = r_3 \times 7.5 \]

Total Distance Calculation:

Kaitlyn aims for a total of 2 miles. Before any calculations, we need to identify when she reaches the distance of 1.25 miles.

Calculations to Reach 1.25 Miles:

Assumed Paces:

1. First Segment (Slow Jog): Let's say \( r_1 = 0.25 \) miles per minute.
   • Distance covered in 5 minutes: \[ d_1 = 0.25 \times 5 = 1.25 \text{ (not applicable in this phase for total 2 miles)} \]

You might want to adjust the speed to 0.1 or an appropriate figure since 0.25 already seems high to reach 1.25 in the slow segment. Let’s calculate correctly.

First Segment Calculation:

Assume a total plan of how far she will get by analyzing not just the speed but actual miles to clear in minutes.

1. To clarify, at 5 minutes running at a slower jog speed might give just a small distance, let's say, in total if:

• \( r_1 = 0.1 \) mile per minute.
• Therefore, in 5 min, she covers \( 0.5 \) miles.

2. Second Segment (Fast Run):

• If it’s \( r_2 = 0.1429 \) (which means actually far more time than calculated).
• In 7 minutes, she'd cover around \( 0.97 \) thus the combination would hint where by tracking exactly that adding would have been hitting that 1.25 miles.

Find the blended metric on break:

• She take time total would be over that break plus her full return to track each segment attributable.

Doing these pieces leads us where she'd stretch time until min adjusted though they add up over break down more directly as before pointed away from 1.25 more as key fill.

Domain Interval Determination:

To find the exact time to reach precisely 1.25 miles:

• Combine minimum total times + break + distance.
• Total is about going into speed with the last intervals bounding exact lengths.

Her complete timing for segments including necessary run distance pushes to \( > 15 \) total divided leading.

1. Summing thus: She would reach \( \text{sum} = 12 (\text{workout time jogging fast}) + 3 \) (break) + the additional or really \(7.5\) thus leaving at 22.5 being likely range over those.

Final Conclusion:

Based on re-checking intervals ensuring mileage correct would thus yield \( 15 \leq s < 22.5 \) for completion of just under load shown for obtaining correctly within total combination pulling.

Thus, the correct option identifying the total time where she definitively hits 1.25 miles is, D. 15 ≤ s ≤ 22.5.

Answer 4

I appreciate very much your assistance!

Answer 5

You're very welcome! I'm glad I could help. If you have any more questions or need further assistance on any topic—whether it's math or something else—feel free to ask. Good luck with your exercises, and have a great day!