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?(1 point)

Question

A) d= 1/8m for t ≤16:d=2 for t >16 but t ≤ 21: d =1/4m -3.25 for t >21 ≤41
B) d= 1/8m for t ≤16:d=2 for t >16 but t ≤ 21
C) d= 1/8m for t ≤16:d=2 for t >16 but t ≤ 21: d =1/8m 20.75 for t >21 ≤41
D) d= 1/8m for t ≤16:d=1/4m -3.25 for t >21 ≤41

Answer 1

To analyze Layla's workout routine and determine which function equation accurately represents the scenario, we need to break down her workout:

1. Warm-up Jog:

   • Distance = 2 miles
   • Time = 16 minutes
   • Rate = Distance/Time = 2 miles / 16 minutes = 1/8 miles per minute

2. Stretching:

   • Time = 5 minutes

3. Running:

   • Distance = 5 miles
   • Time = 20 minutes
   • Rate = Distance/Time = 5 miles / 20 minutes = 1/4 miles per minute

Now we can translate these into a piecewise function:

• For the first segment (light jog):

  • From \( t = 0 \) to \( t = 16 \), the function is \( d = \frac{1}{8}m \).

• For the second segment (stretching):

  • From \( t = 16 \) to \( t = 21 \) (5 minutes of stretching), her distance remains constant at \( d = 2 \).

• For the third segment (running):

  • From \( t = 21 \) to \( t = 41 \) (20 minutes of running). In this case, the distance increases according to the speed of 1/4 miles per minute, which can be represented as:
  • Starting from 2 miles at \( t = 21 \):
    • For \( t \) in this range, we can determine the new distance equation by how it increases from the distance already covered (2 miles) plus how the distance increases according to time.

Given these calculations, let's analyze the answer options:

• Option A:

  • \( d = \frac{1}{8}m \) for \( t \leq 16 \); \( d = 2 \) for \( 16 < t \leq 21 \); \( d = \frac{1}{4}m - 3.25 \) for \( 21 < t \leq 41 \) (not applicable)

• Option B:

  • \( d = \frac{1}{8}m \) for \( t \leq 16 \); \( d = 2 \) for \( 16 < t \leq 21 \) (not including the running part)

• Option C:

  • \( d = \frac{1}{8}m \) for \( t \leq 16 \); \( d = 2 \) for \( 16 < t \leq 21 \); and it recognizes a distance increase for subsequent time, but could be incorrect in specifics.

• Option D:

  • \( d = \frac{1}{8}m \) for \( t \leq 16 \); with the distance being represented incorrectly post-stretching.

The definition of distance and time does not seem perfectly reflected in any option provided.

However, among given options, Option B appears the most consistent logically and mathematically to match the initial running conditions for the two segments of the workout to recognize the stretching period, albeit incomplete.

So, the answer representing the scenario most accurately is B) \( d = \frac{1}{8}m \) for \( t \leq 16 \); \( d = 2 \) for \( 16 < t \leq 21 \).