Video Transcript
If a sphere is inscribed in a
cylinder with a volume of 16𝜋 centimeters cubed and the height of the cylinder is
the same as its diameter, find the volume of the sphere.
To answer this question, we note
first that if the sphere is inscribed in the cylinder, then the sphere is touching
each face of the cylinder without any gaps. This then means that the diameter
of the sphere is the same as the diameter of the cylinder. And since the height of this
cylinder is the same as its diameter, the height of the cylinder also equals the
diameter of the sphere. The radius of both the sphere and
the cylinder is then ℎ over two, which equals half the diameter.
Now let’s remind ourselves of the
formulae for the two volumes we’re concerned with. That’s the volume of a cylinder,
which is 𝜋𝑟 squared ℎ, and the volume of a sphere, which is four over three 𝜋𝑟
cubed. We’re told that the volume of the
cylinder is 16𝜋 centimeters cubed. And we can use this to find the
radius 𝑟 of the cylinder, which is half its height. We can then use this to find the
volume of the sphere. So the volume of the cylinder is
16𝜋 centimeters cubed, which is equal to 𝜋𝑟 squared ℎ.
So now leaving out the units for
the moment, we see we can divide through by 𝜋. And we have 𝑟 squared ℎ is equal
to 16. But now remember that 𝑟 is equal
to ℎ over two. And so multiplying this through by
two gives two 𝑟 is equal to ℎ. Now, substituting this for ℎ in our
equation, we have 𝑟 squared multiplied by two 𝑟 is equal to 16. That is, two 𝑟 cubed equals
16. And dividing both sides by two
leaves us with 𝑟 cubed equal to eight. Taking the cube root on both sides,
we have 𝑟 equal to two.
So now making a little space, we
substitute our value for 𝑟 into the formula for the volume of the sphere to get the
volume equals four over three 𝜋 times two cubed. That’s four over three 𝜋 times
eight. And this evaluates to 32 over three
times 𝜋. Hence, the volume of the sphere
inscribed in the cylinder with volume 16𝜋 centimeters cubed is 32 over three 𝜋
centimeters cubed.
