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Optimization of inscribed circle

  1. Jan 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Given the function
    y = 12- 3x^2,
    find the maximum semi-circular area bounded by the curve and the x-axis.

    2. Relevant equations

    A= Pi(r^2)

    3. The attempt at a solution
    I found my zeros, 2 and -2, and my maximum height of 12 from the y'.

    A' = 2Pi(r)
     
    Last edited: Jan 5, 2010
  2. jcsd
  3. Jan 5, 2010 #2

    vela

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    Plot the function. The answer will be clear.
     
  4. Jan 5, 2010 #3

    Dick

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    Well, no. r isn't equal to 2. If it's an inscribed semicircle bounded by the x-axis the circle must have center (0,0), agree? Draw a picture. It's a wee bit less than 2. So the equation of the circle is r^2=x^2+y^2. If it's inscribed the semicircle must be tangent to the parabola y=12-3*x^2 at the point of intersection. Find dy/dx for each and set them equal. It works out particularly easy if you find dy/dx for the circle using implicit differentiation.
     
  5. Jan 5, 2010 #4

    vela

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    You are, of course, correct. I should have thought about it more carefully.
     
  6. Jan 5, 2010 #5

    Dick

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    You didn't say anything wrong. I wasn't responding to your suggestion. I was responding to the original post. Plotting is always a great idea!
     
    Last edited: Jan 5, 2010
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