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

A right circular cone has a radius of 3π₯ + 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 π and height β is π = 1/3 ππΒ²β.

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Video Transcript

A right circular cone has a radius of three π₯ 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 π and height β is π equals one third ππ squared β.

So weβre told us that the radius was three π₯ plus six and our height is three less than the radius, which we can simplify. So β is equal to three π₯ plus three. So if the volume is equal to one-third ππ squared β, we can plug in three π₯ plus six for π and three π₯ plus three for β.

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 π₯ plus six, which is three π₯ plus six times three π₯ plus six. And now we need a foil. And when foiling that, it gives us nine π₯ squared plus 18π₯ plus 18π₯ plus 36. So letβs simplify that before we multiply by π₯ plus one. And we get π times nine π₯ squared plus 36π₯ plus 36 times π₯ plus one. Now letβs distribute.

Distributing the nine π₯ squared to π₯ plus one, we get nine π₯ cubed plus nine π₯ squared. Distributing the 36π₯ to π₯ plus one, we get 36π₯ squared plus 36π₯. And distributing the 36 to the π₯ plus one, we get 36π₯ plus 36. Now we can combine like terms. Therefore, the volume is equal to π times nine π₯ cubed plus 45π₯ squared plus 72π₯ plus 36.

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