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November 14th, 2007, 01:29 PM   #1
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Circle Problem

Can someone please help me with this problem?

Malcolm is running around a circular track. The track’s radius is 50 meters. Malcolm
runs at a constant speed of 4 meters per second. If he runs for 32 seconds, how far, in a
straight line, will he be from his starting point?
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November 15th, 2007, 03:14 AM   #2
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Well, if the speed v=4m/s and he runs for t=32 second then he runs s=v*t=32*4=128 meters. Now since the radius of the circle is 50 meters, it's circumference is C=2*50*π=100π (well need this later). Imagine the circle and on its perimeter a point that denotes Malcolm's starting point and a second point denoting his end point and connect each point with the circle's center. Now, the angle between them is the ratio of the distance Malcolm ran and the total circumference of the circle multiplied with 2π. That is: θ=128/(100π)*2π=64/25. Now, connect the two points together and that way you get a triangle with two sides of length r and an angle between them - θ. From this it's easy to calculate the remaining triangle side: divide θ in to equal angles with a line connecting the circles center and the unknown triangle's midpoint. That way you can say: sin(θ/2)=x/(2r) -> x =2rsin(θ/2)≈95.8016 meters. Here's a picture for better understanding:

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