Perimeter of Athletic Lanes

JadeKiara · #1 $Old$ July 1st, 2009, 05:17 AM

An athletic track has four lanes. The lanes are 1 metre apart. If the inside lane is exactly 400 m, with 100m straights, find the lengths of the other three lanes.

Can I please have an explanation for obtaining the answer to this problem?

CaptainBlack · #2 $Old$ July 1st, 2009, 05:32 AM

Quote:

Originally Posted by JadeKiara $View Post$

An athletic track has four lanes. The lanes are 1 metre apart. If the inside lane is exactly 400 m, with 100m straights, find the lengths of the other three lanes.

Can I please have an explanation for obtaining the answer to this problem?

Each lane may be represented as two parallel 100m straights with semicircular arcs connecting their ends (drawing a picture is a good idea at this point).

As you are told the length of the inside lane you can work out the radii r of the semicircular sections of that lane.

As the lanes are 1m appart you know the radii of the semicircular sections for the other lanes are r+1, r+2 and r+3 metres. The starights are still 100m long.

Please take it from here.

CB

pickslides · #3 $Old$ July 1st, 2009, 05:35 AM

Consider lane 1, length = 400 m exactly, 100 for each of 2 straights and 100 for each of 2 arch sections

One of the arch sections is $\pi\times r = 100$

therefore $r= \frac{100}{\pi}$ we just have to add 1 radius for each subsequent lane to find its total distance.

Lane 1 = 400m $4\times 100m$

Lane 2 = 100m for each straight with arch section with radius $r = \frac{100}{\pi}+1$

Lane 2 total = $2\times 100 +2\times \pi \left( \frac{100}{\pi}+1\right)= \cdots$

Lane 3 = 100m for each straight with arch section with radius $r = \frac{100}{\pi}+2$

Lane 3 total = $2\times 100 +2\times \pi \left( \frac{100}{\pi}+2\right)=\cdots$

can you find lane 4?

JadeKiara · #4 $Old$ July 2nd, 2009, 03:58 AM

Therefore, Lane 2 is 406.28 m, Lane 3 is 412.57 m, and Lane 4 is 418.85 m. Thank you for all your help! Now I will be an expert on the size of athletic tracks.