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An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track.
a) Draw a figure that visually represents the problem. Let x and y represent the length and width of the rectangular region.
b) Determine the radius of the semicircular ends of the track.
c) Use the result of part b to write an equation in terms of x and y, for the distance traveled in one lap around the track. Solve for x.
d) Use the result of part c to write the area A of the rectangular region as a function of x.

  • Pre-Calculus -

    length of rectangle -- x
    width of rectangle --- y

    b) radius of semicircle is y/2

    c) distance = 2 lengths + 2 halfcircles
    = 2x +2(1/2) π(y/2)^2
    = 2x + π(y^2)/4

    2x + (1/4)πy^2 = 200
    times 4
    8x + πy^2 = 800
    x = (800 - πy^2)/8

    d) -- poorly worded question.
    Since they had you solve for x in c) they should have asked for the area in terms of y , not x

    check your typing.

    BTW, I would have defined the radius of the semicircle as r
    then the width of the rectangle would be 2r, and we can avoid some of these nasty fractions.

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