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.
