Analyzing Olympic Records: Finding k & n in t = kd^n

[JPR]

It is claimed that athletic records follow the law t = kd^n, where d is the distance in metres, t is the time and k and n are constants. The Olympic records (prior to the 22nd games in Moscow, 1980) for certain track events for men, are given in the following table.
Distance 200m 400m 800m 1500m 5000m
Time 19.83 sec 43.86 sec 1m43.5 sec 3m34.9 sec 12m20.4 sec

(a) plot a graph of log10t against log10d and determine whether these values support the above claim.

(b) If this claim is true, in which of the above distances is the record most likely to be broken next?

(c) If the claim is true, find values for k and n.



This is a question on my year 12 Maths B (Australia) assignment. I've got no dramas with the first two questions - I am able to deduce that log10t = 1.1545 log10d - 1.352, and I have found the answer in b to be the 1500m. However for question 3 I can't see how I am able to derive the two constants. It probably has something to do with the equation or there is something simple that I am missing. I don't want any answers but could someone please put me onto the right track.

Any help appreciated.

 

[HallsofIvy]

IF "t = kd^n" is true then log(t)= nlog(d)+ log(k) which means that the graph of log(t) versus log(d) is a straight line (if we let y= log(t), x= log(d), and b= log(k), y= nx+ b). Is that what you saw when you graphed them? Can you find the slope and y intercept of your line? How is k related to the y intercept?

 

[JPR]

Thanks for the help.

Yes the graph of log(t) vs log(d) was a straight line, and gave the formula y=1.1545x - 1.352. The graph also had an R^2 value of 0.9994 which I thought was pretty amazing. I know the values for the slope and y-intercept and given the formula you mentioned (log(t)= nlog(d)+ log(k) which by the way we haven't encountered in class 'yet') I can see how I can find the values for k and n from these numbers. Thanks again.