Question

The ratio of the surface area of two spheres is 3:2. The volume of the larger sphere is 2,916 in^3. What is the volume of the smaller sphere?

Answer 1

The equation for the surface area of a sphere is A = 4 pi r^2.
The equation for the volume of a sphere is V = 4/3 pi R^3.

Say R is the radius of the bigger sphere. Then it has surface area:
4 pi R^2
It has volume:
4/3 pi R^3

Say r is the radius of the smaller sphere. Then it has surface area:
4 pi r^2
It has volume:
4/3 pi r^3

The ratio between the surface areas is then just the surface area of the bigger one divided by the surface area of the smaller one:
(4 pi R^2) / (4 pi r^2) = R^2 / r^2
The question says that this is 3:2, which means that:
R^2 / r^2 = 3 / 2.
We can get rid of the fractions by multiplying diagonally through the equation:
3r^2 = 2R^2

Now the volume of the larger sphere is 2.916. We can plug this into the volume equation to get the radius of the larger sphere, which is R:
V = 4/3 pi R^3
2.916 = 4/3 pi R^3
R^3 = (3 * 2.916) / (4 * pi)
R^3 = 0.69614
R = 0.8863 inches

Now we can plug this into our surface area ratio equation to get r, the radius of the smaller sphere:
3r^2 = 2R^2

Now substitute in R:
3r^2 = 2 * (0.8863)^2
3r^2 = 1.57095
r^2 = 0.52365
r = 0.72364 inches

Now we can use this to calculate the volume for the smaller sphere using the sphere volume equation:
V = 4/3 pi r^3
V = 4/3 pi (0.72364)^3
V = 1.58727 inches^3

Answer 2

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