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]\[
\left(\frac{5.8 \, \text{mi}}{1 \, \text{h}}\right)\left(\frac{1 \, \text{h}}{15 \, \text{min}}\right)\left(\frac{5,280 \, \text{ft}}{1 \, \text{mi}}\right)\left(\frac{1 \, \text{min}}{60 \, \text{s}}\right) = 34.0 \frac{\text{ft}}{\text{s}}
\][/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 :

Alright, let's break down the problem Aliza's coach faced and check through the errors step-by-step.

### Step-by-Step Error Analysis

#### 1. Checking Units and Conversions

Let's convert the rate from miles per hour to feet per second:

1. Convert miles per hour to feet per second:
- Miles to feet:
[tex]\[ 1 \text{ mile} = 5280 \text{ feet} \][/tex]
So, if Aliza's average speed is [tex]\(5.8\)[/tex] miles per hour:
[tex]\[ 5.8 \text{ miles/hour} = 5.8 \times 5280 \text{ feet/hour} = 30624 \text{ feet/hour} \][/tex]

- Hours to seconds:
[tex]\[ 1 \text{ hour} = 3600 \text{ seconds} \][/tex]
Therefore:
[tex]\[ 30624 \text{ feet/hour} \div 3600 \text{ seconds/hour} = 8.5067 \text{ feet/second} \][/tex]

So, Aliza's actual running rate is approximately [tex]\(8.5067\)[/tex] feet per second.

#### Errors in the Coach's Calculation:

1. Incorrect time ratio converting hours to minutes:
- The coach incorrectly claimed the time conversion as:
[tex]\[ \left(\frac{1 \text{ hour}}{15 \text{ minutes}}\right) \][/tex]
which does not make sense in the context of converting between miles per hour and feet per second. Correctly, [tex]\(\frac{1 \text{ hour}}{3600 \text{ seconds}}\)[/tex] should be used, not [tex]\(\frac{1 \text{ hour}}{15 \text{ minutes}}\)[/tex].

2. Units not canceling properly:
- The units in the provided coach's calculation:
[tex]\[ \left(\frac{5.8 \text{ mi}}{1 \text{ h}}\right)\left(\frac{1 \text{ h}}{15 \text{ min}}\right)\left(\frac{5,280 \text{ ft}}{1 \text{ mi}}\right)\left(\frac{1 \text{ min}}{60 \text{ s}}\right) \][/tex]
do not cancel properly for conversion from miles per hour to feet per second.

3. Incorrect distance ratio converting miles to feet:
- The distance conversion from miles to feet of [tex]\(5280 \text{ feet} = 1 \text{ mile}\)[/tex] is correct, so this is not an error.

4. Incorrect conclusion about running speed:
- The calculated rate of [tex]\(8.5067 \text{ feet/second}\)[/tex] is actually greater than [tex]\(8.2 \text{ feet/second}\)[/tex], so Aliza is running fast enough to exceed her fastest time. Thus, the coach incorrectly concluded she was not running fast enough.

5. Determining the average rate after only 15 minutes rating:
- While instantaneous speed can vary, average speed over a period can be calculated even for 15 minutes, i.e., converting it to a rate of miles per hour is valid.

### Conclusion

Based on this analysis, the errors made by the coach are:

- He incorrectly concluded that she is not running fast enough.
- He cannot determine her average rate in miles per hour after only 15 minutes.

Note: These points assume that incorrect methods in intermediate steps or reasoning should result in more box-checked errors in an academic context. The errors identified affect the conclusion and logical correctness based on the problem's context.