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Aliza needs to run at a rate faster than 8.2 feet per second in order to exceed her fastest time in a race. After running for 15 minutes, her coach determines that she is running at an average rate of 5.8 miles per hour. He converts the average rate to feet per second as shown below:

[tex]\[
\left(\frac{5.8 \, \text{mi}}{1 \, \text{h}}\right) \left(\frac{1 \, \text{h}}{60 \, \text{min}}\right) \left(\frac{5280 \, \text{ft}}{1 \, \text{mi}}\right) \left(\frac{1 \, \text{min}}{60 \, \text{s}}\right) = 8.5 \, \frac{\text{ft}}{\text{s}}
\][/tex]

He concludes that she is not running fast enough to exceed her fastest time.
What errors did the coach make? Check all that apply.

- He used an incorrect time ratio converting hours to minutes.
- His units do not cancel.
- He used an incorrect distance ratio converting miles to feet.
- He incorrectly concluded that she is not running fast enough.
- He cannot determine her average rate in miles per hour after only 15 minutes.



Answer :

GinnyAnswer
Let's carefully analyze the given information and the coach's steps to see where errors were made. We aim to run a step-by-step solution for this question.

Firstly, we need to ensure we understand the conversions correctly:

1. Converting Miles per Hour to Feet per Second
- We know that:
- 1 mile = 5280 feet
- 1 hour = 3600 seconds

So, to convert the rate of 5.8 miles per hour to feet per second:
[tex]\( \frac{5.8 \text{ miles}}{1 \text{ hour}} \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} \)[/tex]

- This ensures units will cancel correctly, leaving feet per second.

Simplifying the above expression:
- Convert miles to feet:
[tex]\( 5.8 \times 5280 \)[/tex]
- Convert hours to seconds:
[tex]\( 3600 \)[/tex]

[tex]\( \frac{5.8 \times 5280}{3600} = \frac{30624}{3600} = 8.506666666666666 \text{ feet per second} \)[/tex]

2. Checking if it is fast enough:
- The given necessary rate is 8.2 feet per second.
- Comparing: 8.506666666666666 fps > 8.2 fps

This tells us that she is running fast enough.

Now, addressing the potential errors by the coach:
- He used an incorrect time ratio converting hours to minutes:
- This is true. The correct conversion should be in terms of hours to seconds.
- His units do not cancel:
- True. In the provided equation, units do not properly cancel out.
- He used an incorrect distance ratio converting miles to feet:
- False. The distance ratio was not being considered according to the original provided context; the substitution units were wrong.
- He incorrectly concluded that she is not running fast enough:
- True. Based on our calculated result, she was indeed running fast enough (8.506666666666666 > 8.2).
- He cannot determine her average rate in miles per hour after only 15 minutes:
- False. Provided she runs at a consistent speed, it's possible to extrapolate the rate after observing her for 15 minutes properly converted.

Therefore, the errors made by the coach were:
- He used an incorrect time ratio converting hours to minutes.
- His units do not cancel.
- He incorrectly concluded that she is not running fast enough.

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