solid mensuration

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the perimeter of an ellipse with an area equal to 12pi square units if the non opposite vertices are 5units apart

  • solid mensuration -

  • solid mensuration -

    What a nasty question.
    the area of the ellipse is abπ , where a and b are each 1/2 of the axes

    so abπ = 12π --->ab = 6 or b = 6/a
    let A(a,0) and B(0,b) be the two adjacent vertices
    given AB = 5
    √(a^2 + b^2) = 5
    a^2 + b^2 = 25
    a^2 + 36/a^2 = 25
    a^4 + 36 = 25a^2
    a^4 - 25a^2 + 36 = 0
    a^2 = (25 ± √481)/2 = appr 23.4658 or appr 1.53414
    a = 4.844157.. or a = 1.2386... ignoring the 2 negative values
    then b = 6/a
    if a = 4.844148, then b = 1.2386
    if a = 1.2386 , then a = 4.844157
    ahhh, summetrical answers

    So we basically have the same ellipse, one horizontal, one vertical

    The perimeter of an ellipse is one of the most challenging questions dealing with the ellipse.
    It requires Calculus and finding the length of a curve.
    Here is a link that has some approximation methods

    Here is an applet that let's you find all properties of an ellipse
    enter the 2a= 9.6883
    enter 2b = 2.47721..
    and press calculate to get a perimeter of appr. 20.87

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