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

1 Answer
Feb 23, 2016

Answer:

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)#