Calculus

4. Find the area of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed in the region enclosed by the graphs of f(x) = 18 – x^2 and g(x) = 2x^2 – 9.

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  3. 145
asked by Anonymous
  1. find the derivatives, then plug it in to the orginal questioin.then use the length to find the area as it is an optimization question

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    posted by saphire
  2. not the anti-derivative?

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    posted by Anonymous
  3. let the vertex in quadrant I be (x,y)
    so the base of the rectangle is 2x
    its height is (18-x^2) - (2x^2 - 9)
    = 27 - 3x^2

    Area = 2x(27-3x^2)
    = 54x - 6x^3
    d(Area)/dx = 54 - 18x^2
    = 0 for a max of area

    18x^2 = 54
    x^2 = 3
    x = ±√3
    using x = √3
    area = 54√3 - 18(3)
    = 54(√3-1) or approx 39.53

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    posted by Reiny
  4. can I ask why did you subtract?

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    posted by Anonymous
  5. suppose we take x=1 (within the rectangle)
    What is the height at x=1 ?
    f(1) = 18-1 = 17
    g(1) = 2-9 = -7
    so the actual height of the rectangle when x=1 is 17 - (-7) = 24

    Did I not do a subtraction of f(x) - g(x) ?
    notice by subbing in x=1 into the simplified subtraction answer of 27-3x^2
    I get 27 - 3 = 24

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    posted by Reiny
  6. In response to Reiny, at the end, wouldn't it be "area = (54(sqrt3) -6(sqrt3)^3)? And then you would get an answer of app. 62.354?

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    posted by Anonymous
  7. Why is the base of the rectangle 2x?

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    posted by Anonymous

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