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.
### 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.
