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A right circular cylinder is to be inscribed in a sphere of given radius. Find the ratio of the height to the base radius of the cylinder having the largest lateral area.

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    In my diagram , I arbitrarily let the radius of the cylinder be 1 unit
    let the radius of the cylinder be r , and let the height of the cylinder be 2h
    That way I can say:
    r^2 + h^2 = 1
    r^2 = 1 - h^2 or r = (1- h^2)^(1/2)

    Surface area (SA) = 2πr^2 + 2πrh
    = 2π(1-h^2) + 2π(1-h^2)^(1/2) h
    = 2π(1-h^2) + 2π (h^2 - h^4)^(1/2)

    d(SA)/dh = 2π [-2h + (1/2)(h^2 - h^4)((-1/2) (2h - 4h^3) ]
    = 0 for a max of SA

    (1/2)(2h-4h^3)/√(h^2 - h^4) = 2h
    (1/2)(2h)(1 - 2h^2) / (h√(1-h^2) ) = 2h

    (1 - 2h^2)/√(1 - h^2) = 2h
    1 - 2h^2 = 2h√(1-h^2)
    square both sides
    1 - 4h^2 + 4h^4 = 4h^2 - 4h^4

    8h^4 - 8h^2 + 1 = 0
    solving this I got
    h^2 = (2 ± √2)/4

    case1: h^2 = (2+√2)/4
    h = ..9238796 , r = ..38268.. , h/r = 2.41421.. or 1 + √2

    case2: h^2 = (2-√2)/4
    h = .38268... , r = .9238.. , h/r = .4142 .. or -1 + √2

    Looking at Wolfram
    http://www.wolframalpha.com/input/?i=maximize+2π%281-h%5E2%29+%2B+2π+%28h%5E2+-+h%5E4%29%5E%281%2F2%29

    I will take case 2 as my answer.

    Nasty, nasty question

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