Can you work out the radius of the curved sections in order to produce an accurate scale drawing?
The total length of the inside edge of the inner lane is 400 metres. If you take away the two 85 metre straights you are left with the circumference of the two semicircles added together (230 metres). C = 2pi r so to make the raduis the subject of this formula you rearrange it to make r = C/2pi. So 330 metres divided by 6.283185307179586476925286766559 (2pi) equals 36.605636911135927226843265575678 metres.
Can you work out where each runner should start so that they all run 200m in total?
To find the staggered start you will need to first find the radius of the semicircle. The radius of the first lane (inner lane) semicircle is 36.605636911135927226843265575678 metres. This multiplied by pi (not 2pi because we are working with semicircles (half circles)) gives you the distance of the semicircular part of the 200 metre race (115 metres). This figure, 115 metres plus on 85 metres, then minus 200 metres, gives you how far in from the start of the straight the starting line is; which in this case, is zero metres. So basically the formula to finding how far in the starting line is from the start of the straight is (R x pi + 85) - 200.
For the next lane (2nd from inside), the calculation is the same, except for you must add 1.25 metres to the radius. So,
(37.855636911135927226843265575678 x pi + 85) - 200 = 3.9269908169872415480783042290984
These are the answers for the rest of the lanes..
Lane 3 start @ 7.8539816339744830961566084582m from start of straight.
Lane 4 start @ 11.7809724509617246442349126873m from start of straight.
Lane 5 start @ 15.7079632679489661923132169164m from start of straight.
Lane 6 start @ 19.6349540849362077403915211455m from start of straight.
Lane 7 start @ 23.5619449019234492884698253746m from start of straight.
Lane 8 start @ 27.48893571891069083654812960369m from start of straight.
Can you work out where each runner should start so that they all run 400m in total?
Here, the formula is basically the same as the last question, except this time we are working with full circles, not semicircles.
The distance from finish line = (2pi x r + 170) - 400
Another way to figure the answer out is to simple double the last questions answer.
Answers
Lane 2 start @ 7.8539816339744830961566084581969m from the finish line.
Lane 3 start @ 15.707963267948966192313216916396m from the finish line.
Lane 4 start @ 23.5619449019234492884698253746m from the finish line.
Lane 5 start @ 31.4159265358979323846264338328m from the finish line.
Lane 6 start @ 39.269908169872415480783042291m freom the finish line.
Lane 7 start @ 47.1238898038468985769396507492m from the finish line.
Lane 8 start @ 54.97787143782138167309625920738m from the finish line.