1. ## 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

2. ## 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 $\displaystyle S=124\pi.$