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Real Life Application (3 points): Hook: 13-year old fastest long distance runner! now What You Know. vie joined the cross-country team and is tracking her progress this year. She has recorded her times in the table shown. Answer the following in complete sentences with evidence. Determine if her rate has been proportional Determine her unit rate Predict what her time will be if the next race is 3 miles long. ype your answer here: Miles Minutes 0.5 5 1.5 15 2.0 20 Bonus! (1-point Extra Credit): Write an equation to represent Evie's running pace for any number of minutes, m.

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Real Life Application

Hook: Meet Evie, a remarkable 13-year-old who has become the fastest long-distance runner in her age group! She has joined the cross-country team and is diligently tracking her progress.

Now What You Know: Evie has recorded her running times for different distances in the table below:

| Miles | Minutes | |-------|---------| | 0.5 | 5 | | 1.5 | 15 | | 2.0 | 20 |

  1. Determine if her rate has been proportional.
    To determine if Evie's running rate is proportional, we need to check if the ratio of miles to minutes remains constant. Looking at the table:
  • For 0.5 miles, the rate is \( \frac{0.5 \text{ miles}}{5 \text{ minutes}} = 0.1 \text{ miles per minute} \)
  • For 1.5 miles, the rate is \( \frac{1.5 \text{ miles}}{15 \text{ minutes}} = 0.1 \text{ miles per minute} \)
  • For 2.0 miles, the rate is \( \frac{2.0 \text{ miles}}{20 \text{ minutes}} = 0.1 \text{ miles per minute} \)

Since the ratio of miles to minutes (0.1 miles per minute) is consistent across all recorded distances, we can conclude that Evie's rate is proportional.

  1. Determine her unit rate.
    Evie's unit rate, which represents her speed or pace, is \( 0.1 \text{ miles per minute} \). This means she runs 0.1 miles every single minute.

  2. Predict what her time will be if the next race is 3 miles long.
    To predict Evie's time for a 3-mile race, we can use her unit rate. Knowing that she runs at a pace of 0.1 miles per minute, we find the time it will take for her to complete 3 miles:

\[ \text{Time} = \frac{\text{Distance}}{\text{Rate}} = \frac{3 \text{ miles}}{0.1 \text{ miles per minute}} = 30 \text{ minutes} \]

Thus, if the next race is 3 miles long, we predict that Evie will finish it in 30 minutes.

Bonus! (1-point Extra Credit):

To write an equation that represents Evie's running pace for any number of miles, we can denote the number of miles as \( d \) and the time in minutes as \( t \). Since we know her pace is 0.1 miles per minute, we can express this relationship as:

\[ t = 10d \]

where \( t \) is the time in minutes and \( d \) is the distance in miles. This equation indicates that for every mile, it takes Evie 10 minutes to run.

What's the answers? Can you put them in a box?

Sure! Here are the answers clearly presented in a box:

1. Determine if her rate has been proportional: 
   Yes, Evie's running rate is proportional because the ratio of miles to minutes is consistent at 0.1 miles per minute.

2. Determine her unit rate: 
   Evie's unit rate is 0.1 miles per minute.

3. Predict her time for a 3-mile race: 
   If the next race is 3 miles long, Evie will finish in 30 minutes.

Bonus! Write an equation to represent Evie's running pace:
   The equation is: t = 10d, where t is time in minutes and d is distance in miles.

Feel free to ask if you need any more help!

Write an equation to represent Evie's running pace for any number of minutes, m

To write an equation that represents Evie's running pace for any number of minutes \( m \), we need to relate the distance \( d \) she can run in those minutes.

Since we know that Evie's unit rate is 0.1 miles per minute, we can express distance as a function of time:

\[ d = 0.1m \]

Where:

  • \( d \) is the distance in miles,
  • \( m \) is the time in minutes.

So the equation representing Evie's running pace for any number of minutes \( m \) is:

\[ d = 0.1m \]

This means that for every minute she runs, she covers 0.1 miles.