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Thread: Spheres surfaces maximum

  1. #1
    May 2013

    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
    Super Member
    Jun 2009

    Re: Spheres surfaces maximum

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