Answered

Izumi ran the first 110 yards in 16 seconds.



Part A

How fast, in miles per hour, was Izumi running for the first 110 yards? There are 1,760 yards in a mile. Show or explain all your work.



Part B

Izumi's goal is to run a 4-minute mile. If he maintained the same speed for the first quarter mile, how fast, in miles per hour, will he have to run the final three-quarters of a mile to achieve his goal? Explain how you calculated your answer.

Verified Answer :

  • Part A: Calculate speed by dividing distance (110 yards) by time (16 seconds) converted to miles and hours: \(\frac{110/1760}{16/3600} = \boxed{14.0625 \text{ mph}}\).
  • Part B: Calculate the time taken for the first quarter mile (0.25 miles) at 14.0625 mph (\(0.01777\dots\) hours), subtract it from the total time (\(4/60\) hours) to find the remaining time (\(0.04888\dots\) hours) for the remaining 0.75 miles, and divide distance by this time: \(\boxed{15.3409 \text{ mph}}\).

Explanation

  1. Analyzing Part A: Calculating Current Speed

We analyze Izumi's run to determine his speed and requirements for a 4-minute mile. In Part A, we are given that Izumi ran 110 yards in 16 seconds. To find the speed in miles per hour (mph), we convert the distance to miles and the time to hours. Given that 1,760 yards equal 1 mile, the distance is \(\frac{110}{1760} = 0.0625\) miles. Since there are 3,600 seconds in an hour, the time is \(\frac{16}{3600} \approx 0.00444\) hours. The speed is calculated as \(\text{distance} / \text{time} = \frac{0.0625}{0.00444} = 14.0625\) mph.

  1. Analyzing Part B: Calculating Required Speed for Goal

In Part B, we need to determine the required speed for the remaining three-quarters of a mile to complete a 1-mile run in exactly 4 minutes. A 4-minute mile is equivalent to \(\frac{4}{60} = \frac{1}{15}\) hours. The first quarter-mile (440 yards or 0.25 miles) is covered at the same speed calculated in Part A, which is 14.0625 mph. The time taken to complete the first quarter-mile is \(\frac{0.25 \text{ miles}}{14.0625 \text{ mph}} = 0.01777\dots\) hours. Subtracting this from the total allowed time gives the remaining time for the final three-quarters of a mile: \(\frac{1}{15} - 0.01777\dots \approx 0.04888\dots\) hours. Finally, the required speed for the last 0.75 miles is \(\frac{0.75 \text{ miles}}{0.04888\dots \text{ hours}} \approx 15.3409\) mph.

  1. Final Results

By performing the calculations:

  • Part A: The speed for the first 110 yards is \(\boxed{14.0625 \text{ mph}}\).
  • Part B: The required speed for the final three-quarters of a mile is \(\boxed{15.3409 \text{ mph}}\) (or exactly \(\frac{675}{44}\) mph).

Examples

This type of calculation is common in athletics and logistics. For instance, a marathon runner might calculate their average speed over different segments to ensure they reach their finish time goal. If a runner starts slowly and then picks up their pace for the remaining portion of the race, this math helps determine exactly how fast they need to sprint to compensate for their initial pace. It is a fundamental application of the relation \(\text{distance} = \text{speed} \times \text{time}\).