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Math Help - Spheres surfaces maximum

  1. #1
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    Spheres surfaces maximum

    The distance between centers of two spheres of radii 4cm and 9cm, respectively, is 35cm. How far from the center of the smaller sphere is a point P along the line of centers of the spheres from which the sum of the areas of the visible spherical surfaces is maximum?

    Any kind of help would be appreciated. Thanks
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  2. #2
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    Re: Spheres surfaces maximum

    Taking tangents from the point P to the spheres defines two caps. If P is a distance x from the centre of the smaller sphere, then the depth of the smaller cap will be 4-16/x, and the depth of the larger one 9-81/(35-x).
    The surface area of a spherical cap depth h, sphere radius r is 2\pi rh, so the combined surface area of the two caps will be
    S=2\pi(4)(4-16/x)+2\pi(9)(9-81/(35-x)).
    For maximum S, differentiate this and equate to zero.
    Assuming this to be correct, that gets you x=8 and a corresponding S=124\pi.
    Thanks from HallsofIvy
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