Find, to the nearest tenth, the volume of
a sphere given that the circumference of its great circle is 90𝜋 inches.
So first of all, what do we mean by the
great circle of a sphere? Well, formally, it is the intersection of
the sphere and any plane — that’s a two-dimensional slice — which passes through the center
of the sphere. On our diagram, this is one such great
circle. But in fact, there are infinitely many
depending on the angle of the plane we draw.
Now we know that, in order to find the
volume of any sphere, we use the formula four-thirds 𝜋𝑟 cubed. So in order to answer this question, we
need to calculate the radius of our sphere. We can do this using the information
given about the circumference of the great circle because we know that the circumference of
any circle is found using the formula 𝜋𝑑 or two 𝜋𝑟.
We can therefore form an equation using
the version of the circumference formula that involves 𝑟. Two 𝜋𝑟 is equal to 90𝜋. And we can solve this equation in order
to determine the radius of the sphere. First, we cancel a factor of 𝜋 from each
side, giving two 𝑟 equals 90. And then we divide by two to find that 𝑟
is equal to 45. The units for this will be inches. Finally, we can substitute this value for
the radius into our formula for the volume of a sphere, giving four-thirds multiplied by 𝜋
multiplied by 45 cubed.
As we’re asked to give our answer to the
nearest tenth, it’s reasonable to assume we can use a calculator to help with this
question. So evaluating on a calculator gives
121,500𝜋. Or as a decimal, this is equivalent to
381,703.5074. As we’re rounding to the nearest tenth,
our deciding digit is in the hundredths column. That’s a zero. So we’re rounding down.
So we find that the volume of the sphere
whose great circle has a circumference of 90𝜋 inches, to the nearest tenth, is 381,703.5