I was wondering about the following:

Consider a sphere with radius $ r$ , then the volume equals $ \frac{4}{3}\pi r^3$ . Now consider to cover this sphere with a shell of thickness $ h$ . Then the new volume becomes $ \frac{4}{3}\pi(r+h)^3$ . I was wondering if we can compute the value of $ h$ for which the new sphere has twice the volume of the original sphere. This leads to the following expression

$ $ \frac{4}{3}\pi(r+h)^3 = \frac{8}{3}\pi r^3\ h^3 + 3h^2r + 3hr^2+r^3 = 2 r^3\ h^3 + 3h^2r + 3hr^2-r^3 = 0$ $ This is a cubic polynomial, with coefficients $ (1,3r,3r^2,-r^3)$ . I was wondering if there exists an easy expression for these kind of cubics, for easiness, it may be assumed that $ r$ is integer.

Seeing the origin of this question I am also interested in finding the real root of the cubic. It would be nice if this is an relative easy expression, which does not need for the use of, for example, Cardano’s method.