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 𝜋 𝑟 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 𝜋 𝑟 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
Distributing the nine 𝑥 squared to 𝑥 plus one, we get nine 𝑥 cubed plus nine
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.