Answered

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 :

GinnyAnswer
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.