A right circular cone has a radius
of three 𝑥 plus six and its height is three units less than its radius. Express the volume of the cone as a
polynomial function, knowing that the volume of a cone with radius 𝑟 and height ℎ
is 𝑉 equals one third 𝜋𝑟 squared ℎ.
So we’re told us that the radius
was three 𝑥 plus six and our height is three less than the radius, which we can
simplify. So ℎ is equal to three 𝑥 plus
three. So if the volume is equal to
one-third 𝜋𝑟 squared ℎ, we can plug in three 𝑥 plus six for 𝑟 and three 𝑥 plus
three for ℎ.
Now since we have a one-third, that
can cancel with the threes for the height, and this will make our work a little bit
easier. So now we need to square three 𝑥
plus six, which is three 𝑥 plus six times three 𝑥 plus six. And now we need a foil. And when foiling that, it gives us
nine 𝑥 squared plus 18𝑥 plus 18𝑥 plus 36. So let’s simplify that before we
multiply by 𝑥 plus one. And we get 𝜋 times nine 𝑥 squared
plus 36𝑥 plus 36 times 𝑥 plus one. Now let’s distribute.
Distributing the nine 𝑥 squared to
𝑥 plus one, we get nine 𝑥 cubed plus nine 𝑥 squared. Distributing the 36𝑥 to 𝑥 plus
one, we get 36𝑥 squared plus 36𝑥. And distributing the 36 to the 𝑥
plus one, we get 36𝑥 plus 36. Now we can combine like terms. Therefore, the volume is equal to
𝜋 times nine 𝑥 cubed plus 45𝑥 squared plus 72𝑥 plus 36.