Question 335398: You are in charge of constructing a fence around a running track. The fence is to be built around the track so that there is a uniform gap between the outside edge of the track and the fence. What is the maximum width of the gap between the track and the fence if no more than 630 meters of fencing is used? Hint: use the equation for the circumference of a circle, C=2(pi)r, to help you. The track is an oval with the straight length 100 m and the width 81 m.
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! Typically, running tracks are a rectangle with two half-circles attached at each end.
For the two sides of the rectangle that do not have the half-circles on the end, 200 meters of the fence will be used. This leaves 430 meters to be used for the circles.
Since the width of the track is 81 m, the radius of each half-circle is 81/2 m. However, if the gap is x meters in width, the result will be a circle with radius 81/2 + x.
To maximize the gap, we will need to use all of the fence. Thus, we have:
2π(81/2 + x) = 430 (the circumference of each half-circle is C = πr)
==> π(81 + 2x) = 430
==> 2x + 81 = 430/π
==> 2x = 430/π - 81
==> x = 215/π - 81/2 ≈ 28 m.
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