Can Women beat Men in the Marathon?

In 1964, the world record time for the men’s marathon was 1 hour, 15 minutes and 34 seconds faster than the world record time for the women’s marathon. In 1985, the difference was only 13 minutes and 54 seconds. Are women catching up?

The following table shows data for the world record times in the women’s and men’s marathon.

 

Men’s record times

 

Women’s record times

Date (Years since 1900)

Time
(hours)

Name

Country

 

Date (Years since 1900)

Time
(hours)

Name

Country

9.0027

2.8792

Robert Fowler

USA

 

64.3918

3.4625

Dale Greig

GBR

9.1178

2.7811

James Clark

USA

 

64.5534

3.3258

Mildred Sampson

NZE

13.3616

2.6378

Harry Green

GBR

 

67.3452

3.2561

Maureen Wilton

CAN

13.4137

2.6017

Alexis Ahlgren

SWE

 

67.4904

3.1239

Anni Pede

FRG

20.6137

2.5431

Hannes Kolehmainen

FIN

 

70.4904

3.0481

Caroline Walker

USA

25.7808

2.4836

Albert Michelson

USA

 

71.3534

3.0283

Elizabeth Bonner

USA

35.2466

2.4636

Fushashige Suzuki

JPN

 

71.6658

2.7750

Adrienne Beame

AUS

35.2548

2.4456

Yashuo Ikenaka

JPN

 

74.8219

2.7733

Chantal Langlace

FRA

35.8411

2.4450

Kitei Son

JPN

 

74.9178

2.7317

Jacqueline Hansen

USA

47.2986

2.4275

Yun Bok Suh

KOR

 

75.3041

2.7117

Liane Winter

FRG

52.4521

2.3450

Jim Peters

GBR

 

75.3370

2.6708

Christa Vahlensieck

FRG

53.4493

2.3111

Jim Peters

GBR

 

75.7808

2.6386

Jacqueline Hansen

USA

53.7589

2.3094

Jim Peters

GBR

 

77.3315

2.5875

Chantal Langlace

FRA

54.4849

2.2942

Jim Peters

GBR

 

77.6932

2.5797

Christa Vahlensieck

FRG

58.6466

2.2547

Sergey Popov

SOV

 

78.8082

2.5414

Grete Waitz

NOR

60.6932

2.2544

Abebe Bikila

ETH

 

79.8055

2.4589

Grete Waitz

NOR

63.1315

2.2542

Toru Terasawa

JPN

 

80.8192

2.4281

Grete Waitz

NOR

63.4548

2.2411

Leonard Edelen

USA

 

81.8247

2.4281

Allison Roe

NZE

64.4493

2.2319

Basil Heatley

GBR

 

83.2959

2.4247

Grete Waitz

NOR

64.8055

2.2031

Abebe Bikila

ETH

 

85.2959

2.3783

Joan Benoit

USA

65.4466

2.1833

Morio Shigematsu

JPN

 

85.3041

2.3503

Ingrid Kristiansen

NOR

67.9233

2.1600

Derek Clayton

AUS

 

 

 

 

 

69.4110

2.1428

Derek Clayton

AUS

 

 

 

 

 

81.9315

2.1383

Rob de Castella

AUS

 

 

 

 

 

84.8055

2.1347

Steve Jones

GBR

 

 

 

 

 

85.3014

2.1200

Carlos Lopes

POR

 

 

 

 

 

88.2932

2.1139

Belayneh Dinsamo

ETH

 

 

 

 

 

 

 

 

  1. Plot the data in the above table on an appropriate set of axes
  2. The data in the above table can be modeled using expressions in the form

T = aebY + c

where T is the record time, in hours, Y is the year after 1900 and a, b, and c are real constants.

Using the highlighted data points in the above table, set up simultaneous equations to solve for the constants a, b and c and hence determine expressions for T, in terms of Y, for both men and women.

  1. Superimpose sketches of the graphs of the expressions for T over your data plot.
  2. The data points selected to set up the simultaneous equations to solve for the constants a, b and c included the endpoints for both men and women. Comment on the choice of these endpoints in terms of the validity of your expressions as models of future world record times.
  3. In terms of exponential functions and the modeling of future world record marathon times, comment on the significance of the constant, c.
  4. Explain at which point in your analysis it first becomes clear that women will in fact beat men in the marathon at some stage in the future.
  5. Determine the point of intersection between the graphs of your expressions for the world record marathon times for men and women. Show this point on your graph.
  6. According to your analysis, what date and for which world record time will women equal, and, thereafter, beat, men in the marathon?
  7. In general terms, discuss the validity of your analysis and the limitations of your models as predictors of future world record marathon times.

 

 


 

Can Women beat Men in the Marathon?

Suggested solutions

  • Italicised text in red indicate specific objectives and notes for teachers as well as appropriate marks for a marking scheme
 
  1. The following data plot has been generated by MS Excel.

Alternatively, the data plot can be generated by a graphing calculator (STAT PLOT) as the following screen dumps show. T is represented by the pronumeral y and Y is represented by x.

     

  • The window screen dump is important here because the scale and range and domain are not apparent on the plot screen alone.
  •  

  • Accurate plot of data (1 mark)
  • Data plot appropriately labeled (1 mark)
  • Y as the independent variable (1 mark)

 

2. The shaded data for men can be written as the three coordinates, (9.0027, 2.8792),
(65.4466, 2.1817), (88.2932, 2.1139).

Substituting these values into the general equation yields three equations:

2.8792 = ae9.0027b + c (equ. 1)

2.1817 = ae65.4466b + c (equ. 2)

2.1139 = ae88.29322b + c (equ. 3)

  • Appropriate set of simultaneous equations (1 mark)

 

Arranging these equations slightly gives us

ae9.0027b = 2.8792 - c (equ. 1)

ae65.4466b = 2.1817 - c (equ. 2)

ae88.29322b = 2.1139 - c (equ. 3)

Dividing equ. 2 by equ.1 gives us

(equ. 4)

And dividing 3 by 2 gives us

(equ. 5)

We can equate equ. 4 and equ. 5 by using a fundamental index law to give us

which, in turn, gives us

  • Appropriate analytical elimination of constants (1 mark)
  • Note that there are other acceptable methods for eliminating b and solving for c.
 

The above equation is solved using the "zero" function on a graphing calculator.

A graph of the function expressed by the left hand side of the above equation is sketched as is shown in the following two screen dumps. c is represented by the pronumeral x.

  • Note that the window should not be arrived at by guesswork altogether. Students should anticipate a zero of between 2 and 2.5, given the current world record time.
 

Left and right bounds and a guess are entered into the zero function of the calculator and a zero value is returned as follows:

Thus the solution is given by

C = 2.053718

  • Appropriate application of technology to solve equations (1 mark)
  • Note that although the solution is accurate to six decimal places, it still represents an approximation. Hence an "approximately equal to" symbol rather than an "equal to" symbol is used.
  •  

  • There are other appropriate forms of technology that students can use here, such as graphing software and computer algebra systems. The important point to consider when technology is used, in this case, is that students have demonstrated the consideration of domain constraints.
  •  

 

Substituting c = 2.053718 into equation 4 gives us

Substituting this into equation 1 gives us

  • Appropriate substitution of values to determine a, b and c (1 mark)
 

Since the data is given accurate to four decimal places, a, b and c are similarly rounded. Hence, the model for the men’s marathon can be expressed as,

Tm = 1.1113e-0.033Y + 2.0537, Y" 9.0027

  • Appropriate expression for T (1 mark)

 

The shaded data for women can be written as the three coordinates, (75.3370, 2.6679), (77.3315, 2.5846), (85.3041, 2.3500).

Substituting these values into the general equation yields three equations:

2.6679 = ae75.370b + c (equ. 1)

2.5846 = ae77.3315b + c (equ. 2)

2.3500 = ae85.3041b + c (equ. 3)

  • Appropriate set of simultaneous equations (1 mark)

Arranging these equations slightly gives us

ae75.370b = 2.6679 - c (equ. 1)

ae77.3315b = 2.5846 - c (equ. 2)

ae85.3041b = 2.3500 - c (equ. 3)

Dividing equ. 2 by equ. 1 gives us

(equ. 4)

And dividing 3 by 2 gives us

(equ. 5)

We can equate equ. 4 and equ. 5 to give us

which, in turn, gives us

  • Appropriate analytical elimination of constants (1 mark)
  • Note that there are other acceptable methods for eliminating b and solving for c.
 

This equation is again solved using the "zero" function on a graphing calculator resulting in the following screen dump:

Thus, the solution is given by c = 2.0523991

  • Appropriate application of technology to solve equations (1 mark)
  • Note that although the solution is accurate to six decimal places, it still represents an approximation. Hence an "approximately equal to" symbol rather than an "equal to" symbol is used.
  •  

  • There are other appropriate forms of technology that students can use here, such as graphing software and computer algebra systems. The important point to consider when technology is used, in this case, is that students have demonstrated the consideration of domain constraints.
  •  

 

Substituting this into equation 4 gives us

Substituting this into equation 1 gives us

  • Appropriate substitution of values to determine a, b and c (1 mark)
 

Since the data is given accurate to four decimal places, a, b and c are similarly rounded. Hence, the model for the women’s marathon can be expressed as,

Tw = 149.5102e-0.0729Y + 2.0523,

  • Appropriate expression for T (1 mark)
  1. The following sketches have been generated by MS Excel.

  Correct sketches of graphs (1 mark)

  • Appropriate domain and range of graphs (1 mark)

 

Alternatively, the sketches can be generated by a graphing calculator as the following screen dumps show.

  • Note the restricted domain and range of both men’s and women’s modeled data. It is not straightforward to display graphs of functions with restricted domains on graphing calculators, but nonetheless important.
 

4. A reason for selecting the last data point is that the model should, at the very least, predict a record equal to or better than the current record. Predictions of future marathon times slower than the current record don’t make sense in real world applications.

  • Appropriate comments (1 mark)

5. The significance of c is

  • it is the asymptote of the exponential function. As Y approaches infinity, T approaches the asymptote, c.
  • in marathon terms, c defines the limit of human potential - men or women will run no faster than the time given by the asymptote.
  • Appropriate comments (1 mark)
 

6. The women’s asymptote of 2.5024 hours is less than the men’s of 2.0537 hours suggesting that eventually women’s marathon times will be faster than men’s. It is at this point, the calculation of both asymptotes, that it first becomes clear that women will in fact beat men in the marathon at some time in the future.

  • Appropriate explanation (1 mark)

7. To see whether Tm and Tw intersect, they are equated to solve for Y.

1.1113e-0.033Y + 2.0537 = 149.5102e-0.0729Y + 2.0524

1.1113e-0.033Y - 149.5102e-0.0729Y + 0.0013 = 0

This equation is solved using the zero function on a graphing calculator resulting in the following screen dumps.

Thus, the solution is given by

Y Ù 121.29708 years after 1900

When Y = 121.29708,

Tm = 1.11129858e-0.033025319 • 121.29708 + 2.053718

™ 2.073952918 hours

Ù 2.0740 hours

  • The unrounded values of a, b and c are used here since the accuracy the answer is otherwise adversely affected.
 

This point of intersection is shown in MS Excel on the graph in the solution to Question 3.

Alternatively, the point of intersection can be found using the "intersect" function on a graphing calculator as the following screen dump shows:

The point of intersection is given as (121.32137, 2.0739367).

Since the zero of above was given for T = and not exactly

T = 0, the point of intersection of (121.32137, 2.0739367) will be used.

  • Appropriate application of technology to determine points of intersection (1 mark)
  • Correct point of intersection (1 mark)
  • Correct labeling of point of intersection (1 mark) — see graph in Q.3.

 

  1. In real terms, the values of years after 1900 and T =

hours can be expressed as follows

121.32137 + 1900 = 2021.32137

= 0.32137 365 days after January 1 in 2021

= 117.30005 days after January 1 in 2021

= April 27, 2021

2. 0739367 hours = 2 hours and 0. 0739367 60 minutes

= 2 hours and 4.436202 minutes

= 2 hours, 4 minutes and 26.17 seconds

  • Again, unrounded values are used for accuracy.
 

Thus, the world record women’s times are predicted to equal, and thereafter better, those of men on the 27th of April in the year 2021 in a time of 2 hours, 4 minutes and 26.17 seconds.

  • Appropriate expression of solution in real terms (1 mark)
 
  1. Some discussion points might include
  • The superimposition of the graphs of the model over the data plots indicate good fits by eye and so the models appear "reasonable". The graphs do also indicate, however, that the men’s model is not particularly effective between 1900 and 1936 and then again between 1947 and 1964.
  • The accuracy of the model as a predictor can be "measured" appropriately a number of ways, one of which being the calculation of the average absolute difference between actual data points and the values predicted by the models. The average difference, in this case, for men is 5.33% and 4.61% for women, which clearly shows a good fit.
  • We cannot tell, however, that the differences between real and modeled data are minimised as they would be with regression analyses. The accuracy of the model in fitting the data is largely dependent on the choice of the three data points. As they were already given, there is no way of being certain about their appropriateness.
  • New world records not consistent with the model will significantly affect future modeling, particularly since the most recent record is always selected as a data point.
  • Small rounding errors, or varying rounding techniques, significantly effect the accuracy of model, if the actual date is required. The model is, however, quite effective, if all that is required is the year in which women equal men.
  • Appropriate discussion (1 mark)

 

Criteria Mapping for VCE Mathematical Methods Unit 4 Analysis Task:

 

 

Outcomes

 

1

 

 

2

 

 

3

 

 

Criteria

 

1

2

3

1

2

3

1

2

 

Specific Objectives

Marks

 

 

 

 

 

 

 

 

Q. 1.

Accurate plot of data

1

 

 

¸

 

 

 

 

¸

 

Data plot axes appropriately labelled

1

¸

 

 

 

 

 

 

 

 

Years since 1900 as the independent
variable

1

 

 

 

¸

 

 

 

 

Q. 2.

Men

Appropriate set of simultaneous equations

1

¸

¸

 

 

¸

 

 

 

 

Appropriate analytical elimination of
constants

1

 

 

¸

 

¸

 

 

 

 

Appropriate application of technology to
solve equations

1

 

 

 

 

 

 

¸

¸

 

Correct values for a, b and c

1

 

 

¸

 

 

 

 

¸

 

Appropriate expression for T (values for a, b
and c accurate to 4 decimal places)

1

 

 

 

¸

 

 

¸

 

Women

Appropriate set of simultaneous equations

1

¸

¸

 

 

¸

 

 

 

 

Appropriate analytical elimination of
constants

1

 

 

¸

 

¸

 

 

 

 

Appropriate application of technology to
solve equations

1

 

 

 

 

 

 

¸

¸

 

Correct values for a, b and c

1

 

 

¸

 

 

 

 

¸

 

Appropriate expression for T (values for a, b
and c accurate to 4 decimal places)

1

 

 

 

¸

 

 

¸

 

Q. 3.

Correct sketches of graphs

1

¸

 

 

 

 

 

¸

¸

 

Appropriate domain and range of graph

1

¸

 

 

¸

 

 

 

 

Q. 4.

Appropriate comments on the choice of
endpoints

1

 

 

 

 

 

¸

 

 

Q. 5.

Appropriate comments on the significance
of c

1

 

¸

 

¸

 

¸

 

 

Q. 6.

Appropriate explanation of when it
becomes clear women will beat men

1

 

 

 

 

 

¸

 

 

Q. 7.

Appropriate application of technology to
determine points of intersection

1

 

 

 

 

¸

 

¸

 

 

Correct point of intersection

1

¸

 

¸

 

 

 

¸

 

 

Correct labelling of point of intersection on
graph

1

¸

 

 

¸

 

 

 

 

Q. 8.

Appropriate expression of solution in real
terms

1

 

 

 

¸

 

 

 

 

Q. 9.

Appropriate discussion of validity and
limitations of model

1

 

 

 

¸

 

¸

 

 

 

Total marks

23

 

 

 

 

 

 

 

 

Rate this item
(0 votes)

Email This email address is being protected from spambots. You need JavaScript enabled to view it.
Go to top