Aliza needs to run at a rate faster than 8.8 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:
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.
The Correct Answer and Explanation is:
Correct Answers:
- ✅ 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.
Explanation
In this problem, Aliza’s coach attempts to determine whether she is running faster than 8.8 feet per second (ft/s) by converting her average speed of 5.8 miles per hour (mph) into ft/s. However, his unit conversion contains a critical time conversion error, which leads to a faulty conclusion.
Let’s examine the coach’s setup:5.8 mi/h×(1 h15 min)×(5280 ft1 mi)×(1 min60 s)5.8 \text{ mi/h} \times \left(\frac{1 \text{ h}}{15 \text{ min}}\right) \times \left(\frac{5280 \text{ ft}}{1 \text{ mi}}\right) \times \left(\frac{1 \text{ min}}{60 \text{ s}}\right)5.8 mi/h×(15 min1 h)×(1 mi5280 ft)×(60 s1 min)
Error 1: Incorrect Time Conversion
The term 1 h15 min\frac{1 \text{ h}}{15 \text{ min}}15 min1 h is incorrect. Since 1 hour = 60 minutes, this should be reversed. If converting miles per hour to feet per second, the time conversions should use the correct unit ratios:
- 1 hour=60 minutes=3600 seconds1 \text{ hour} = 60 \text{ minutes} = 3600 \text{ seconds}1 hour=60 minutes=3600 seconds
Thus, the correct conversion would be:
5.8 mi/h×(5280 ft1 mi)×(1 h3600 s)5.8 \text{ mi/h} \times \left(\frac{5280 \text{ ft}}{1 \text{ mi}}\right) \times \left(\frac{1 \text{ h}}{3600 \text{ s}}\right)5.8 mi/h×(1 mi5280 ft)×(3600 s1 h)
This results in:5.8×5280/3600≈8.5 ft/s5.8 \times 5280 / 3600 \approx 8.5 \text{ ft/s}5.8×5280/3600≈8.5 ft/s
Error 2: Units Do Not Cancel Correctly
Because the time conversion was incorrect, the units do not cancel properly, resulting in a meaningless unit (not feet per second). This shows a fundamental error in dimensional analysis.
Error 3: Incorrect Conclusion
After correcting the math, we find that Aliza is running at approximately 8.5 ft/s, which is very close to the target of 8.8 ft/s. Given slight variations, this may be fast enough, and his conclusion is premature.
Final Note
The coach can determine her average speed after 15 minutes (since average speed is total distance/time), so that statement is incorrect.
