Question

If the volume of a sphere doubles, what is the ratio of the surface area of the new, larger sphere to the old?

Answer 1

The ratio of the surface area of the new, larger sphere to the old is
`root(3)(4)`

Explanation:

Let's start with two formulas - for surface area of a sphere `S` and for its volume `V`, assuming the radius of a sphere is `R`:
`S=4piR^2`
`V=4/3piR^3`

To double the volume, we have to increase the radius by multiplying it by `root(3)(2)`.
Indeed, let `R_1 = Rroot(3)(2)`
Then the volume of a sphere with radius `R_1` will be
`V_1 = 4/3piR_1^3 = 4/3 pi (Rroot(3)(2))^3 = 8/3piR^3` - twice larger than original volume.

With radius `R_1 = Rroot(3)(2)` the surface area of a new sphere will be
`4 pi R_1^2 = 4 pi R^2(root(3)(2))^2=4 root(3)(4) pi R^2`

The ratio of the new surface area to the old one equals to
`(4 root(3)(4) pi R^2) / (4 pi R^2) = root(3)(4)`