Question

the height of a cone is twice the radius of its base. what expression represents the volume of the cone, in cubic units? \\(\frac{2}{3}\pi x^3\\) \\(\frac{4}{3}\pi x^2\\) \\(2\pi x^3\\) \\(4\pi x^3\\)

Explanation:

Step1: Recall cone volume formula

The volume $V$ of a cone is given by $V = \frac{1}{3}\pi r^2 h$, where $r$ is the base radius and $h$ is the height.

Step2: Identify given values

From the diagram, $r = x$ and $h = 2x$.

Step3: Substitute values into formula

Substitute $r=x$ and $h=2x$ into the volume formula:
$$V = \frac{1}{3}\pi (x)^2 (2x)$$

Step4: Simplify the expression

First calculate the product of the variables: $(x)^2(2x) = 2x^3$. Then multiply by $\frac{1}{3}\pi$:
$$V = \frac{2}{3}\pi x^3$$

Answer:

$\frac{2}{3}\pi x^3$ (corresponding to the first option)