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:
$\left(\frac{5.8 mi}{1 h}\right)\left(\frac{1 h}{15 min}\right)\left(\frac{5,280 ft}{1 mi}\right)\left(\frac{1 min}{60 s}\right)=34.0 \frac{ft}{s}$
He concludes that she is not running fast enough to exceed her fastest time.
What errors did the coach make? Check all that apply.
A. He used an incorrect time ratio converting hours to minutes.
B. His units do not cancel.
C. He used an incorrect distance ratio converting miles to feet.
D. He incorrectly concluded that she is not running fast enough.
E. He cannot determine her average rate in miles per hour after only 15 minutes.
Answer (2)
The coach used an incorrect time ratio converting hours to minutes.
The coach incorrectly concluded that Aliza is not running fast enough.
The correct speed is approximately 8.51 s f t , which is faster than the required 8.2 s f t .
Therefore, the coach made errors in both the conversion and the conclusion.
Explanation
Analyzing the Conversion Let's analyze the coach's conversion to identify any errors. The coach wants to convert 5.8 miles per hour to feet per second. The correct conversion should be:
5.8 h mi × 1 mi 5280 f t × 3600 s 1 h = 3600 5.8 × 5280 s f t
Calculating the Correct Speed Let's calculate the correct value:
3600 5.8 × 5280 = 8.506666... ≈ 8.51 s f t
Identifying the First Error Now, let's examine the coach's calculation:
( 1 h 5.8 mi ) ( 15 min 1 h ) ( 1 mi 5 , 280 f t ) ( 60 s 1 min ) = 34.0 s f t
We can see that the coach incorrectly used 15 minutes in the conversion instead of 1 hour (60 minutes). The correct time ratio should be 60 min 1 h , but he used 15 min 1 h . This is the first error.
Identifying the Second Error The coach's calculation is incorrect because he divided by 15 minutes instead of using the correct conversion factor of 60 minutes in an hour. The correct calculation should be:
( 1 h 5.8 mi ) ( 60 min 1 h ) ( 1 mi 5280 f t ) ( 60 s 1 min ) = 8.51 s f t
Since Aliza needs to run faster than 8.2 feet per second, and her actual speed is approximately 8.51 feet per second, she is running fast enough to exceed her fastest time. Therefore, the coach incorrectly concluded that she is not running fast enough. This is the second error.
Final Answer The coach made two errors:
He used an incorrect time ratio converting hours to minutes.
He incorrectly concluded that she is not running fast enough.
The other options are incorrect because his units do cancel, he used the correct distance ratio, and he can determine her average rate after 15 minutes.
Examples
Understanding unit conversions is crucial in many real-life scenarios. For instance, when planning a road trip, you might need to convert kilometers to miles to match your car's speedometer or the road signs. Similarly, chefs often convert grams to ounces when following recipes from different countries. Accurate conversions ensure you're using the right measurements, whether it's for distance, volume, or weight, preventing errors and ensuring successful outcomes.
The coach made errors by using an incorrect time ratio in his conversion and incorrectly concluded that Aliza is not running fast enough. The correct speed after conversion is approximately 8.51 feet per second, which exceeds the required 8.2 feet per second. Therefore, Aliza is indeed fast enough in her run.
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