Question

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: \\(\left(\frac{5.8\text{ mi}}{1\text{ h}}\
ight)\left(\frac{1\text{ h}}{15\text{ min}}\
ight)\left(\frac{5280\text{ ft}}{1\text{ mi}}\
ight)\left(\frac{1\text{ min}}{60\text{ s}}\
ight)\approx 34.0\frac{\text{ft}}{\text{s}}\\) he concludes that she is not running fast enough to exceed her fastest time. what errors did the coach make? check all that apply. \\(\square\\) he used an incorrect time ratio converting hours to minutes. \\(\square\\) his units do not cancel. \\(\square\\) he used an incorrect distance ratio converting miles to feet. \\(\square\\) he incorrectly concluded that she is not running fast enough. \\(\square\\) he cannot determine her average rate in miles per hour after only 15 minutes.

Explanation:

Step1: Analyze time conversion ratio

The coach uses $\frac{1\ \text{h}}{15\ \text{min}}$ to convert hours to minutes, but the correct ratio should be $\frac{1\ \text{h}}{60\ \text{min}}$ (or $\frac{60\ \text{min}}{1\ \text{h}}$ for unit cancellation). This is an incorrect time ratio.

Step2: Check unit cancellation

The units in the coach's calculation: $\frac{\text{mi}}{\text{h}} \times \frac{\text{h}}{\text{min}} \times \frac{\text{ft}}{\text{mi}} \times \frac{\text{min}}{\text{s}}$. The units $\text{mi}$, $\text{h}$, $\text{min}$ do cancel out to leave $\frac{\text{ft}}{\text{s}}$, so this is not an error.

Step3: Analyze distance conversion ratio

The coach uses $\frac{5280\ \text{ft}}{1\ \text{mi}}$, which is the correct ratio for converting miles to feet, so this is not an error.

Step4: Calculate correct speed

First, convert 5.8 miles per hour to feet per second:
$$5.8\ \frac{\text{mi}}{\text{h}} \times \frac{5280\ \text{ft}}{1\ \text{mi}} \times \frac{1\ \text{h}}{3600\ \text{s}} = 5.8 \times \frac{5280}{3600}\ \frac{\text{ft}}{\text{s}} = 5.8 \times 1.4667 \approx 8.51\ \frac{\text{ft}}{\text{s}}$$
This speed is faster than 8.2 ft/s, so the coach incorrectly concluded she is not fast enough.

Step5: Evaluate rate validity after 15 min

Average rate can be calculated over any time interval, so determining an average rate after 15 minutes is valid. This is not an error.

Answer:

• He used an incorrect time ratio converting hours to minutes.
• He incorrectly concluded that she is not running fast enough.