### Video Transcript

A sphere of diameter 12 centimetres
is dropped into a right circular cylindrical vessel partly filled with water. If the sphere is completely
submerged in water, the water level in the cylindrical vessel rises by three and
five-ninths of a centimetre. Find the diameter of the
cylindrical vessel.

Let’s begin by calculating the
volume of the sphere. The volume of a sphere with a
radius 𝑟 is given by the formula four-thirds multiplied by 𝜋 multiplied by 𝑟
cubed. Since the sphere has a diameter of
12 centimetres, its radius can be found by halving 12. Half of 12 is six centimetres. Substituting what we know into the
formula, we get that the volume of our sphere is four-thirds multiplied by 𝜋
multiplied by six cubed. Six cubed is six multiplied by six
multiplied by six. And we can divide six by three to
get two.

That leaves us with four times two
times six times six multiplied by 𝜋. Six multiplied by six is 36. We can then multiply 36 by four by
doubling it once and then doubling it again. 36 doubled is 72. Timesing it by two again, and we
get 144. Finally, we can multiply this by
two to get us 288. The volume of the sphere is 288𝜋
centimetres cubed.

The formula for volume of a
cylinder is 𝜋𝑟 squared ℎ, where 𝑟 is the radius of the cylinder and ℎ is its
height. We don’t know the volume. But we do know that the change in
its height is three and five-ninths of a centimetre. The displacement represents the
volume of the sphere. So we can substitute three and
five-ninths into the formula for volume of the cylinder and equate it to 288𝜋 which
we calculated as being the volume of the sphere.

Before we do any further
calculations, we should convert three and five-ninths into an improper fraction. To do that, we first multiply three
by nine which tells us the total number of ninths in three wholes. Three multiplied by nine is 27. So we have twenty-seven ninths and
an additional five-ninths. That gives us a total of thirty-two
ninths. So our equation becomes 288𝜋
equals 𝜋 times radius squared times thirty-two ninths. We can divide through by 𝜋. And we get 288 equals thirty-two
ninths multiplied by 𝑟 squared.

Next, we can divide through by
32. 32 is a factor of 288. But if we haven’t spotted that, we
could’ve repeatedly divided both sides of this equation by two. In fact, 288 divided by 32 is
nine. Notice, 32 multiplied by 10 is
320. 288 is one less 32 than this. So we know that nine is equal to
one-ninth of 𝑟 squared. Next, we’ll multiply both sides of
this equation by nine, which gives us 𝑟 squared is equal to 81.

To find the value of 𝑟, we’ll
square root everything. The square root of 81 is nine. And we should always consider that
there’s actually two solutions whenever we square root. We get a positive and a
negative. However, the radius represents a
length. So it must be a positive value
since the length can never be negative. We calculated the radius of the
vessel to be nine centimetres. The diameter is double the
radius. Nine multiplied by two is 18. So the diameter is 18
centimetres.