Answered

Aliza needs to run faster than 8.2 feet per second to exceed her fastest time in a race. After running for 15 minutes, her coach determines 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}}{15 \, \text{min}}\right)\left(\frac{5,280 \, \text{ft}}{1 \, \text{mi}}\right)\left(\frac{1 \, \text{min}}{60 \, \text{s}}\right)=34.0 \, \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 analyze the situation and the steps involved in solving this problem to identify any errors made by the coach.

1. Given Information:
- Aliza needs to run faster than 8.2 feet per second.
- She ran for 15 minutes.
- Her coach determined her average speed was 5.8 miles per hour.

2. Converting 5.8 miles per hour to feet per second:
- First, let's recall the relevant conversion factors:
- 1 hour = 60 minutes
- 1 mile = 5,280 feet
- 1 minute = 60 seconds

3. Step-by-step conversion:
- Convert miles per hour to feet per second:
[tex]\[ \text{Conversion formula:} \quad \left(\frac{5.8 \text{ miles}}{1 \text{ hour}}\right) \left(\frac{5,280 \text{ feet}}{1 \text{ mile}}\right) \left(\frac{1 \text{ hour}}{60 \text{ minutes}}\right) \left(\frac{1 \text{ minute}}{60 \text{ seconds}}\right) \][/tex]

4. Checking the conversions:
- The correct process to convert 5.8 miles per hour to feet per second should be:
[tex]\[ \frac{5.8 \text{ miles}}{\text{hour}} \times \frac{5,280 \text{ feet}}{\text{mile}} \times \frac{1 \text{ hour}}{3,600 \text{ seconds}} = \frac{5.8 \times 5,280 \text{ feet}}{3,600 \text{ seconds}} \approx 8.53 \text{ feet per second} \][/tex]

5. Analyzing the Accuracy and Errors:
- Incorrect conclusion: The coach's final conclusion was incorrect because he calculated 34.0 feet per second instead of the correct value, which is about 8.53 feet per second.
- Incorrect conversion ratio: The coach made a mistake converting miles per hour to feet per second. He didn't appear to fully reduce the time units properly, leading to an incorrect cancellation of units.
- Units not canceling correctly: There is a mistake in unit cancellation in the coach's conversion method that results in an incorrect answer.
- Incorrect distance and time ratios: The coach used incorrect conversions in his calculation rather than the precise method outlined above.
- Average rate in miles per hour: It is valid to determine an average rate in miles per hour after 15 minutes because average speed can be calculated over any period.

In summary, the errors the coach made are:
- He used an incorrect conversion ratio when converting miles per hour to feet per second.
- His units did not cancel out correctly.
- He incorrectly concluded that she is not running fast enough.

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