Find the radius of a sphere whose volume
is nine over two 𝜋 cubic centimeters.
In this problem, we’ve been given the
volume of a sphere and asked to work backwards to determine its radius. Let’s begin by recalling the formula we
use for calculating the volume of a sphere. It’s this, four-thirds 𝜋𝑟 cubed, where
𝑟 represents the radius of the sphere.
Now as we’ve been given the volume and we
know the general formula for working it out, we can set these two values or expressions
equal to one another to give an equation. We have four-thirds 𝜋𝑟 cubed equals
nine over two 𝜋. And in order to determine the radius of
the sphere, we simply need to solve this equation.
First, we notice that there’s a factor of
𝜋 on each side of the equation. So we can cancel this. Or we can think of this as dividing
through by 𝜋, to give four-thirds 𝑟 cubed equals nine over two. Next, we need to divide each side of the
equation by four-thirds in order to leave 𝑟 cubed on its own on the left-hand side.
We recall that dividing by a fraction is
equivalent to multiplying by the reciprocal of that fraction. So to divide by four-thirds, we can
multiply each side of the equation by three-quarters. Doing so will eliminate the factor of
four-thirds on the left-hand side, leaving just 𝑟 cubed. And on the right-hand side, we have nine
over two multiplied by three over four. We multiply the numerators, giving 27,
and multiply the denominators, giving eight. So we find that 𝑟 cubed is equal to 27
over eight. To find the value of 𝑟, we need to
perform the inverse or opposite of cubing, which is cube rooting. So we find that 𝑟 is equal to the cubed
root of 27 over eight.
Now at this point, we remember that, in
order to find the cubed root of a fraction, we can find the cubed root of the numerator over
the cubed root of the denominator. So we have that 𝑟 equals the cubed root
of 27 over the cubed root of eight. And these are both integer values. The cubed root of 27 is three, and the
cubed root of eight is two. So we find that the radius of this sphere
is three over two or 1.5. And as the units for the volume were
cubic centimeters, the units for the radius will be centimeters.
Now of course, we weren’t asked for it in
this problem. But if we needed to calculate the
diameter of the sphere, we just need to recall that the diameter is twice the radius. So if the radius is three over two
centimeters, then the diameter is twice this. The diameter of the sphere is three
centimeters. We’ve completed the problem there. The radius of the sphere whose volume is
nine over two 𝜋 cubic centimeters is three over two centimeters.