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? _≤ x ≤ _

Answer 1

To determine the interval during which Kaitlyn is at 1.25 miles, we first need to break down her running segments and calculate her speed for each segment.

1. Slow jog for 5 minutes:
   Let's assume her slow jog speed is \( v_1 \) miles per minute. The distance covered in this time is:
   \[ \text{Distance} = v_1 \times 5 \]

2. Increased speed for 7 minutes:
   Let's assume her faster speed is \( v_2 \) miles per minute. The distance covered here is:
   \[ \text{Distance} = v_2 \times 7 \]

3. Running for 7.5 minutes:
   Let's assume her speed at this stage is \( v_3 \) miles per minute. The distance covered here is:
   \[ \text{Distance} = v_3 \times 7.5 \]

Now, we know that the total distance she ran is 2 miles. Therefore, we can set up the equation: \[ v_1 \times 5 + v_2 \times 7 + v_3 \times 7.5 = 2 \]

However, to answer the specific question of the interval when she is at 1.25 miles, let's consider when she reaches that distance.

• Calculate the distance at each segment:
  • After her slow jog (5 minutes), the distance is \( v_1 \times 5 \).
  • After increasing her pace for 7 minutes, the distance is: \[ \text{Total Distance after 12 minutes} = v_1 \times 5 + v_2 \times 7 \]
  • After another 7.5 minutes, the total distance will reach 2 miles.

To find when she reaches 1.25 miles, we can set up inequalities for the segments:

1. After 5 minutes: If \( v_1 = \frac{2}{t} \) where \( t \) is total time until she reaches 2 miles.
2. After 12 minutes: Add the distance from the first leg to the second leg and set to 1.25 miles.

Without specific speeds, we cannot literally calculate distances, thus let's assume \( v_1 \) and \( v_2 \) lead up to 1.25 miles shortly after the second leg:

Since \( 1.25 \) is halfway to 2 miles:

• She reaches \( 1.25 \) miles before finishing either the increased speed run or the final segment (roughly halfway through that).

Total running time is 19.5 minutes (5 + 7 + 7.5) + 3 minutes break.
By proportion: Using average speeds over intervals, you'd be estimating around half to two-thirds of total time would indicate she may reach 1.25 miles around 12-15 minutes after starting (after the boost of 7).

So our interval can end up around: 0 ≤ x ≤ 15 Here clarified based on interpretations.