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 
Name 
Country 

Date (Years since 1900) 
Time 
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 





 Plot the data in the above table on an appropriate set of axes
 The data in the above table can be modeled using expressions in the form
T = ae^{bY} + 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.
 Superimpose sketches of the graphs of the expressions for T over your data plot.
 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.
 In terms of exponential functions and the modeling of future world record marathon times, comment on the significance of the constant, c.
 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.
 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.
 According to your analysis, what date and for which world record time will women equal, and, thereafter, beat, men in the marathon?
 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
 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 = ae^{9.0027b} + c (equ. 1)
2.1817 = ae^{65.4466b} + c (equ. 2)
2.1139 = ae^{88.29322b} + c (equ. 3)
 Appropriate set of simultaneous equations (1 mark)
Arranging these equations slightly gives us
ae^{9.0027b} = 2.8792  c (equ. 1)
ae^{65.4466b} = 2.1817  c (equ. 2)
ae^{88.29322b} = 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,
T_{m }= 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 = ae^{75.370b} + c (equ. 1)
2.5846 = ae^{77.3315b} + c (equ. 2)
2.3500 = ae^{85.3041b} + c (equ. 3)
 Appropriate set of simultaneous equations (1 mark)
Arranging these equations slightly gives us
ae^{75.370b} = 2.6679  c (equ. 1)
ae^{77.3315b} = 2.5846  c (equ. 2)
ae^{85.3041b} = 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,
T_{w} = 149.5102e^{0.0729Y} + 2.0523,
 Appropriate expression for T (1 mark)
 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 T_{m} and T_{w} 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,
T_{m }= 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.
 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 27^{th} 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)
 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 
1 



¸ 




Q. 2. Men 
Appropriate set of simultaneous equations 
1 
¸ 
¸ 


¸ 




Appropriate analytical elimination of 
1 


¸ 

¸ 




Appropriate application of technology to 
1 






¸ 
¸ 

Correct values for a, b and c 
1 


¸ 




¸ 

Appropriate expression for T (values for a, b 
1 



¸ 


¸ 

Women 
Appropriate set of simultaneous equations 
1 
¸ 
¸ 


¸ 




Appropriate analytical elimination of 
1 


¸ 

¸ 




Appropriate application of technology to 
1 






¸ 
¸ 

Correct values for a, b and c 
1 


¸ 




¸ 

Appropriate expression for T (values for a, b 
1 



¸ 


¸ 

Q. 3. 
Correct sketches of graphs 
1 
¸ 





¸ 
¸ 

Appropriate domain and range of graph 
1 
¸ 


¸ 




Q. 4. 
Appropriate comments on the choice of 
1 





¸ 


Q. 5. 
Appropriate comments on the significance 
1 

¸ 

¸ 

¸ 


Q. 6. 
Appropriate explanation of when it 
1 





¸ 


Q. 7. 
Appropriate application of technology to 
1 




¸ 

¸ 


Correct point of intersection 
1 
¸ 

¸ 



¸ 


Correct labelling of point of intersection on 
1 
¸ 


¸ 




Q. 8. 
Appropriate expression of solution in real 
1 



¸ 




Q. 9. 
Appropriate discussion of validity and 
1 



¸ 

¸ 



Total marks 
23 







