Question

Layla has a specific workout routine when she trains for track season. She warms up with a light jog for 2 miles at a constant rate for 16 minutes. Then she stretches for 5 minutes before running 5 miles at a constant rate for 20 minutes. Which function equation represents this scenario?

Answer 1

To represent Layla's workout routine with a function equation, we can break it down into two segments: the warm-up jog and the main run.

1. **Warm-up Jog:**
- Distance: 2 miles
- Time: 16 minutes

The pace (rate) for the warm-up jog can be calculated as:
\[
\text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{2 \text{ miles}}{16 \text{ minutes}} = \frac{1}{8} \text{ miles per minute}
\]

2. **Stretch:**
- Time: 5 minutes

This does not involve any distance, so we can just note that there is a 5-minute stretch period.

3. **Main Run:**
- Distance: 5 miles
- Time: 20 minutes

The pace for the main run can be calculated as:
\[
\text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{5 \text{ miles}}{20 \text{ minutes}} = \frac{1}{4} \text{ miles per minute}
\]

Given this breakdown, we can create a piecewise function that includes these periods. Let \( t \) represent time in minutes.

The piecewise function can be described as follows:

\[
d(t) =
\begin{cases}
\frac{1}{8} t & \text{for } 0 \leq t \leq 16 \\
\text{(stretched, distance is constant)} & \text{for } 16 < t \leq 21 \\
\frac{1}{4}(t - 21) + 2 & \text{for } 21 < t \leq 41
\end{cases}
\]

- For the first segment (warm-up jog), the distance \( d \) is \(\frac{1}{8} t\) from \( t = 0 \) to \( t = 16 \).
- For the second segment (stretch), the distance remains constant until \( t \) is 21 minutes, maintaining the distance of 2 miles.
- For the third segment (main run), we need to account for the total distance covered. Here, \( d \) starts from 2 miles and adds \(\frac{1}{4} (t - 21)\) for \( t \) between 21 and 41.

In total, the function captures Layla's workout schedule and distance over time.