Question

#3.) A large running track is constructed to have straight sections into semicircular sections with dimensions given and figure 12.82. Assume that runners always run on the outside of their revolution around the track plus an extra portion of stripes in Macon to end up at the finish line. What should the distance x between the two starting blocks be in order to make it a fair race? Explain your reasoning.

Answer 1

in this question, we will equal the total distance to be travelled by the runners to the same for a fair race.

calculate the path distance for the runner who will run in inside path/ track.

so for path 1 the total distance is,

`D_1=2(\pi r_1+100)+x+d`

here the r1 is the radius that is 45m and d is the distance between the finish line and the starting block 1.

Now, we will calculate for the outside track,

`D_2=2(\pi r_2+100)+d`

here, r2 is 50 m.

for the fair race, D1 and D2 should be equal to each other,

equate both,

`\begin{gathered} D_1=D_2 \\ 2(\pi r_1+100)+x+d=2(\pi r_2+100)+d \end{gathered}`

put the values and solve further,

`\begin{gathered} 2\pi*45+200+x=2\pi*50+200 \\ x=2\pi(50-45) \\ x=2\pi*5 \end{gathered}`

x=2*3.14*5

x=31.4 m

So, the value of x should be 31.4 m for a fair race.