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Video: Multiplying Polynomials to Form a Polynomial Function Involving the Volume of a Cone and Its Dimensions

Bethani Gasparine

A right circular cone has a radius of 3𝑥 + 6 and its height is 3 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 = 1/3 𝜋𝑟²ℎ.

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Video Transcript

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 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.