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Geometry/Area Problem

  1. Dec 20, 2009 #1
    1. The problem statement, all variables and given/known data
    A rectangle is bounded by the x-axis and the semicircle y=√(36-x^2) (see figure). Write the area A of the rectangle as a function of x.

    I will try to explain the figure as best as possible. The figure is basically a semicircle on the Cartesian plane with a domain of [-6,6]. There is a point on the semicircle of (X,Y) where y is the height of the rectangle and x is the maximum width of the rectangle. The value of x is around five on the graph, while the value for y is around 3.5 on the graph (note: they are just estimates to help visualize the figure better).

    2. Relevant equations

    3. The attempt at a solution
    I substitute the (x,y) for (w,h). I then realized that the height stays constant, therefore I can write the height as √(36-w^2), where w is a constant. Now, I do not know how to write the width because the x has a limited domain.
  2. jcsd
  3. Dec 20, 2009 #2
    Couldn't you use a definite integral to evaluate this?
  4. Dec 20, 2009 #3
    Not really. The question is found in a precalculus textbook.
  5. Dec 20, 2009 #4


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    The rectangle with a corner at (x,y) on your circle in the first quadrant has a base of length 2x and its height is y = sqrt(36-x2). Multiply those together and you will have the area as a function of x. Apparently you are not asking for the dimensions of such a rectangle with largest area, just the expression for A as a function of x.
  6. Dec 20, 2009 #5
    Thanks LCKurtz
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