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Sagot :
Let's analyze the problem and find out where the coach went wrong.
1. Conversion of Average Rate from Miles per Hour to Feet per Second:
Given:
- Aliza's average running rate: 5.8 miles per hour (mi/h)
Conversion factors required:
- There are 5,280 feet in a mile.
- There are 60 minutes in an hour.
- There are 60 seconds in a minute.
Formula to convert mi/h to ft/s:
[tex]\[ \left(\frac{\text{miles}}{\text{hour}}\right) \times \left(\frac{\text{feet}}{\text{mile}}\right) \times \left(\frac{\text{hour}}{\text{minutes}}\right) \times \left(\frac{\text{minutes}}{\text{seconds}}\right) \][/tex]
Plugging in the values:
[tex]\[ \left(\frac{5.8 \text{ mi }}{1 \text{ h}}\right) \times \left(\frac{5,280 \text{ ft }}{1 \text{ mi }}\right) \times \left(\frac{1 \text{ h}}{60 \text{ min}}\right) \times \left(\frac{1 \text{ min }}{60 \text{ s}}\right) \][/tex]
Simplify the expression:
[tex]\[ \left(5.8 \times 5280 \div (60 \times 60)\right) \text{ ft/s} \][/tex]
Upon calculating, the average running rate in feet per second is approximately 8.5067 ft/s.
2. Determine if Aliza is running fast enough:
The required speed to exceed her fastest time: 8.2 ft/s.
Comparing the calculated speed:
[tex]\[ 8.5067 \text{ ft/s} > 8.2 \text{ ft/s} \][/tex]
Thus, Aliza is indeed running fast enough to exceed her fastest time.
3. Errors in the Coach's Calculation:
- He used an incorrect time ratio converting hours to minutes: This statement is true as the coach used the incorrect conversion of hours to minutes with [tex]\(\left(\frac{1 h}{15 min}\right)\)[/tex], which is not part of the correct conversion factors.
- His units do not cancel: This is true. The way the coach arranged the units resulted in uncancelled units that led to an incorrect calculation.
- He used an incorrect distance ratio converting miles to feet: This is false. The coach correctly used 5,280 feet per mile.
- He incorrectly concluded that she is not running fast enough: This is true. Based on the correct conversion and calculation, Aliza is indeed running faster than 8.2 feet per second.
- He cannot determine her average rate in miles per hour after only 15 minutes: This is false. The average rate can be determined after any duration, as long as the total distance and time are known.
Thus, the errors made by the coach are:
- 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.
1. Conversion of Average Rate from Miles per Hour to Feet per Second:
Given:
- Aliza's average running rate: 5.8 miles per hour (mi/h)
Conversion factors required:
- There are 5,280 feet in a mile.
- There are 60 minutes in an hour.
- There are 60 seconds in a minute.
Formula to convert mi/h to ft/s:
[tex]\[ \left(\frac{\text{miles}}{\text{hour}}\right) \times \left(\frac{\text{feet}}{\text{mile}}\right) \times \left(\frac{\text{hour}}{\text{minutes}}\right) \times \left(\frac{\text{minutes}}{\text{seconds}}\right) \][/tex]
Plugging in the values:
[tex]\[ \left(\frac{5.8 \text{ mi }}{1 \text{ h}}\right) \times \left(\frac{5,280 \text{ ft }}{1 \text{ mi }}\right) \times \left(\frac{1 \text{ h}}{60 \text{ min}}\right) \times \left(\frac{1 \text{ min }}{60 \text{ s}}\right) \][/tex]
Simplify the expression:
[tex]\[ \left(5.8 \times 5280 \div (60 \times 60)\right) \text{ ft/s} \][/tex]
Upon calculating, the average running rate in feet per second is approximately 8.5067 ft/s.
2. Determine if Aliza is running fast enough:
The required speed to exceed her fastest time: 8.2 ft/s.
Comparing the calculated speed:
[tex]\[ 8.5067 \text{ ft/s} > 8.2 \text{ ft/s} \][/tex]
Thus, Aliza is indeed running fast enough to exceed her fastest time.
3. Errors in the Coach's Calculation:
- He used an incorrect time ratio converting hours to minutes: This statement is true as the coach used the incorrect conversion of hours to minutes with [tex]\(\left(\frac{1 h}{15 min}\right)\)[/tex], which is not part of the correct conversion factors.
- His units do not cancel: This is true. The way the coach arranged the units resulted in uncancelled units that led to an incorrect calculation.
- He used an incorrect distance ratio converting miles to feet: This is false. The coach correctly used 5,280 feet per mile.
- He incorrectly concluded that she is not running fast enough: This is true. Based on the correct conversion and calculation, Aliza is indeed running faster than 8.2 feet per second.
- He cannot determine her average rate in miles per hour after only 15 minutes: This is false. The average rate can be determined after any duration, as long as the total distance and time are known.
Thus, the errors made by the coach are:
- 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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