Below is a table of the men's metric world track records as of August 1976 [Source: Guinness Book of World Records]. At that time, the mile was still a major track event contested by the world's best runners.
| Metric World Records (August 1976) x meters
| T secs
| Year set
| 100
| 9.96
| 68
| 200
| 19.81
| 71
| 400
| 43.9
| 68
| 800
| 103.5
| 76
| 1,000
| 133.9
| 74
| 1,500
| 212.2
| 74
| 2,000
| 291.4
| 76
| 3,000
| 455.2
| 74
| 5,000
| 793.0
| 72
| 10,000
| 1,650.8
| 73
| 20,000
| 3,444.2
| 76
| 25,000
| 4,456.8
| 75
| 30,000
| 5,490.4
| 70
| | ||
If you were an analyst back in 1976 and you had this data, could you use mathematics to make a prediction giving an accurate estimate of what the mile record would be in August 1976?
In this activity, based on "Estimating the Mile Record" [Peter Taylor, Calculus: The Analysis of Functions, Wall & Emerson], you will use increasingly sophisticated mathematical analysis to make better and better estimates. In conclusion, you will compare your estimates to the actual mile record for 1976 and see how your predictions measure up. Note: A mile equals about 1,609.344 meters.
First Analysis.
Note that since a mile is more than 1,500 meters, and less than 2,000 meters, the mile record should be between the records for these two distances. Based on this elementary analysis, fill out the table below:
| Your Lower Bound for mile record: | ____________________ seconds. |
| Your Upper Bound for mile record: | ____________________ seconds. |
| The size of your "Uncertainty Gap": | ____________________ seconds. |
Can you improve on the preceding using data from several races?
(Scratchwork):
| Your Lower Bound for mile record: | ____________________ seconds. |
| Your Upper Bound for mile record: | ____________________ seconds. |
| The size of your "Uncertainty Gap": | ____________________ seconds. |
To improve your estimate (which means looking for a lower Upper Bound and a higher Lower Bound, making your Uncertainty Gap smaller), consider the average speeds.
Compute the average speed for the 1,500 m world's record:
Average Speed = distance/time = __________/__________ = (approximately) __________.
If the mile were run at this speed how many seconds would it take? (Recall that time = distance/speed). Mile record estimate based on average speed for 1,500 m:
Average Speed = distance/time = __________/__________ = (approximately) __________.
If the mile were run at this speed how many seconds would it take? Mile record estimate based on average speed for 2,000 m:
Based on the observation that the greater the distance run, the lower the athlete's average speed, fill out the following chart.
| Your Lower Bound for mile record: | ____________________ seconds. |
| Your Upper Bound for mile record: | ____________________ seconds. |
| The size of your "Uncertainty Gap": | ____________________ seconds. |
To further improve your estimate, you need to analyze the data more
carefully. In the space below, carefully plot the coordinates of the
world's records for the 100, 200, 400, 800, 1,000, 1,500, 2,000,
3,000, and 5,000 meter events. Each record gives a point (distance,
time) in the x-T plane, where x is measured in meters and T in
seconds.
A careful eye reveals that this point graph is slightly concave-up.
To confirm this arithmetically, consider the 400, 800, and 1,000
points, and show that the graph is concave up over this triplet of
points. (Hint: You may want to consider either the straight line
segment connecting the 400 and 1,000 points, or compare the straight
line segment for 400 and 800 with the segment for 800 and 1,000).
Since the graph is concave up, the secant (straight line segment) between the 1,500 and 2,000 points lies above the actual graph, so the actual mile record should be less than the value this secant line has at 1 mile = 1609.344 meters.
Compute the slope of the secant line:__________.
Now give an equation for this line:______________________________.
Plugging in x = 1609.344, your new Upper Bound for the mile record
becomes: __________.
To get a similar new Lower Bound, consider the secant between the
1,000 and 1,500 points.
Explain why the heights of this secant at 1,609.334 gives a lower
bound for the mile record:
Compute the slope of the [1,000, 1,500] secant line:__________.
Now give an equation for this line:______________________________.
Plugging in x = 1609.344, your new Lower Bound for the mile record becomes: __________.
To see how sharp your estimates have now become, fill out the table below:
| Your Lower Bound for mile record: | ____________________ seconds. |
| Your Upper Bound for mile record: | ____________________ seconds. |
| The size of your "Uncertainty Gap": | ____________________ seconds. |
Do considerations of concavity make a big difference in how narrow the range is for your lower/upper bound estimates? __________. Would you suspect that the actual mile record lies inside your interval?
The mile record, as of August 1976, was set by John Walker of New
Zealand at Gothenburg, Sweden, on August 12, 1975, and was 3 minutes
and 49.4 seconds. Translating this into one number (in seconds), did
this world record lie in your sharpest (narrowest) estimated interval?