The current mens running world records are given below Expla

The current men\'s running world records are given below. Explain why a linear model between distance and time, T = mD + c, would be inappropriate. It is proposed that a more suitable model is one which uses a power law, T = a times D^p where a and p are constants. By taking logarithms of both sides of this equation, find the linear relationship between log(T) and log (D).

Solution

(a) As we can see that from above data that as we increase the distance our time taken by athelets get increased more as compared to distance ( Because no can maintain high speed for longer time )

Let\'s us take first two data and plug into the equation

9.58 = 100m + C

19.19 = 200m +C

m = 0.0961 anc C = -0.03

Consider one case of Zersenay Tadsese who run 20000 in 3321

Now we want to calcualte time with above mathematica model

T = 20000x0.0961 - 0.03

T = 1922-0.03 = 1921.97

But the data show it is 3321

Hence this linear model T = mD +C is inappropriate

(b)

T = a x D^p

where a and p are constant

Now taking log on both side

ln T = ln ( a x D^p)

ln T = ln a + ln D^p

ln T = ln a + p ln D

ln T = p lnD + ln a

The above would be linear relationship between ln T and ln D

where a and p are constant

Answer