Problem 24 - Precalculus

Question

Have you or a friend ever run in a \(10 \mathrm{K}(10,000 \text { meter })\) race? When the author polled his precalculus class at UCLA in Fall \(1997,\) he found that there were five students in the class (of 160 ) who said they had run a \(10 \mathrm{K}\) in under 50 minutes. Of those five, two (one male, one female) said they had run a \(10 \mathrm{K}\) in under 40 minutes. The world record for this event is well under 30 minutes. In this exercise you'll look at some of the world records in this event over the past decade. (a) The table that follows lists the world records in the (men's) 10,000 meter race as of the end of the years \(1993,1995,\) and \(1997 .\) After converting the times into seconds, plot the three points corresponding to these records in a coordinate system similar to the one shown. $$\begin{array}{lll}\text { Year } & \text { Time } & \text { Runner } \\\\\hline 1993 & 26: 58.38 & \text { Yobes Ondieki (Kenya) } \\\1995 & 26: 43.53 & \text { Haile Gebrselassie (Kenya) } \\\1997 & 26: 27.85 & \text { Paul Tergat (Kenya) } \\\\\hline\end{array}$$ (GRAPH CAN'T COPY) (b) Use the midpoint formula and the data for 1993 and 1995 to compute an estimate for what the world record might have been by the end of \(1994 .\) Then compute the percentage error (rounded to two decimal places), given that the record at the end of 1994 was 26: 52.23 (set by William Seigei of Kenya). Was your estimate too high or too low? (c) Use the midpoint formula and the data for 1995 and 1997 to compute an estimate for what the world record might have been by the end of \(1996 .\) Then compute the percentage error given that the record at the end of 1996 was 26: 38.08 (set by Salah Hissou of Morocco). Was your estimate too high or too low? Is the percentage error more or less than that obtained in part (b)? (d) Using a coordinate system similar to the one shown in part (a), or using a photocopy, plot the points corresponding to the (actual, not estimated) world records for the years \(1993,1994,1995,1996,1997,\) and 1998 Except for \(1998,\) all the records have been given above. The world record at the end of 1998 was 26: 22.75 (set by Haile Gebrselassie of Kenya). Use the picture you obtain to say whether or not the record times seem to be leveling off.

Step-by-Step Solution

Answer

The estimates for 1994 and 1996 were too low; times do appear to level off slightly by 1998.

Step 1: Convert Times to Seconds

First, convert the race times given for the years 1993, 1995, and 1997 to seconds. For 1993: \(26:58.38 = 26 \times 60 + 58.38 = 1618.38\) seconds.For 1995: \(26:43.53 = 26 \times 60 + 43.53 = 1603.53\) seconds.For 1997: \(26:27.85 = 26 \times 60 + 27.85 = 1587.85\) seconds.

Step 2: Plot Points on Coordinate System

Plot the points (1993, 1618.38), (1995, 1603.53), and (1997, 1587.85) in a coordinate system with the year on the x-axis and time in seconds on the y-axis.

Step 3: Calculate Midpoint for 1994 Estimate

Use the midpoint formula for the years 1993 and 1995 to find an estimate for 1994. \(x = \frac{1993 + 1995}{2} = 1994\) (year)\(y = \frac{1618.38 + 1603.53}{2} = 1610.955\) (seconds)

Step 4: Calculate Percentage Error for 1994

The actual 1994 record time is 26:52.23 = 1612.23 seconds. Calculate percentage error:\[\text{Percentage Error} = \frac{|1610.955 - 1612.23|}{1612.23} \times 100 = 0.079\%\]The estimate was too low.

Step 5: Calculate Midpoint for 1996 Estimate

Use the midpoint formula for the years 1995 and 1997 to estimate for 1996:\(x = \frac{1995 + 1997}{2} = 1996\) (year)\(y = \frac{1603.53 + 1587.85}{2} = 1595.69\) (seconds)

Step 6: Calculate Percentage Error for 1996

The actual 1996 record time is 26:38.08 = 1598.08 seconds. Calculate the percentage error:\[\text{Percentage Error} = \frac{|1595.69 - 1598.08|}{1598.08} \times 100 = 0.15\%\]The estimate was too low. The percentage error is greater than that obtained in part (b).

Step 7: Plot Records from 1993 to 1998

Plot the actual record points: (1993, 1618.38), (1994, 1612.23), (1995, 1603.53), (1996, 1598.08), (1997, 1587.85), (1998, 1582.75). Observe the trend to determine if times appear to level off.

Key Concepts

Midpoint Formula

The midpoint formula is a helpful tool in precalculus and geometry. It is used to find the middle point between two coordinate points in a plane. This is particularly useful when attempting to estimate the value or position of something that lies between two known quantities. The formula to find the midpoint is given by:

• \((x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points between which you want to find the midpoint. After substituting the values, you simply take the average of the x-coordinates and the average of the y-coordinates.
This exercise used the midpoint formula to estimate the world record for a 10K race. By using the times from 1993 and 1995, we calculated what the record might have been in 1994. Similarly, an estimate for 1996 was found using the data from 1995 and 1997. This demonstrates how the midpoint formula can provide a straightforward estimate of changes over time.

Coordinate System Plotting

Plotting points on a coordinate system is a fundamental skill in mathematics and science. It involves placing points on a graph where the horizontal axis represents one variable, such as time, and the vertical axis represents another variable, such as race times in this exercise.
To correctly plot points, follow these steps:

• Identify the x-coordinate (such as the year in this exercise) and the y-coordinate (such as time in seconds).
• Place the point on the graph at the intersection of these coordinates.

In this exercise, the points corresponding to the world records for the years 1993, 1995, and 1997 were plotted.
By plotting these points in a coordinate system, students can visualize the decrease in world record times over these years. It also allows for an evaluation of whether these times appeared to level off over this period. Observing trends in a plotted graph can give insights into patterns and help make predictions.

Percentage Error Calculation

Calculating the percentage error is a way to determine the accuracy of an estimate in comparison to the actual value. It measures how much the estimate deviates from the true value, expressed as a percentage of the actual value. The formula to calculate percentage error is:

• \[\text{Percentage Error} = \left(\frac{|\text{Estimated Value} - \text{Actual Value}|}{\text{Actual Value}} \times 100\right)\% \]

This formula involves the following steps:

• Calculate the absolute difference between the estimated and actual values.
• Divide by the actual value to find the error relative to the actual value.
• Multiply by 100 to convert this number to a percentage.

In the exercise, percentage error was calculated for the estimated world records against the actual records for the years 1994 and 1996. The results show how close the estimates were to the real records, indicated by a percentage. A lower percentage error indicates a more accurate estimate, while a higher percentage suggests less accuracy.