Question 251474: We just blew some air into a spherical balloon and doubled its volume. By how much did we multiply the surface area ? Answer by Theo(13342) (Show Source):
if we solve for r in the equation for the volume of the sphere, we get:
take the cube root of both sides of this equation to get:
if substitute for the value of r in the equation for the surface area of the sphere, then we get:
becomes:
if we double v, then this equation becomes:
if we divide s[2] by s[1] we get:
this simplifies to:
which is the same as:
multiply both sides of this equation by and you get:
for example:
let the volume of the sphere be equal to 300.
we solve for the radius to get:
radius of the sphere equals 4.152830592
we confirm by substituting in the equation for the volume of the sphere to get:
v = 4/3 * pi * (4.152830592)^3 = 300 cubic units.
the surface area of the sphere is equal to 4 * pi * r^2 = 216.7196518 square units.
if the sphere doubles in volume, then the surface area of the new sphere should be equal to 216.7196518 * making the surface area of the new sphere equal to
square units
to confirm this is correct, we need to solve for the radius of the new sphere.
the volume of the new sphere is equal to 600 cubic units.
the radius of the new sphere is calculated to be 5.23223868 units.
the surface area of the new sphere is equal to 4 * pi * r^2 which becomes
