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‘Ÿ and height β„Ž is 𝑉 = 1/3 πœ‹π‘ŸΒ²β„Ž.

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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‘Ÿ and height β„Ž is 𝑉 equals one third πœ‹π‘Ÿ squared β„Ž.

So we’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„Ž is equal to three π‘₯ plus three. So if the volume is equal to one-third πœ‹π‘Ÿ squared β„Ž, we can plug in three π‘₯ plus six for π‘Ÿ and three π‘₯ plus three for β„Ž.

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 let’s simplify that before we multiply by π‘₯ plus one. And we get πœ‹ times nine π‘₯ squared plus 36π‘₯ plus 36 times π‘₯ plus one. Now let’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œ‹ times nine π‘₯ cubed plus 45π‘₯ squared plus 72π‘₯ plus 36.

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