Question

a cylinder with a base diameter of x units has a volume of πx³ cubic units. which statements about the cylinder are true? select two options. the radius of the cylinder is 2x units. the area of the cylinders base is 1/4πx² square units. the area of the cylinders base is 1/2πx² square units. the height of the cylinder is 2x units. the height of the cylinder is 4x units.

Explanation:

Step1: Find the radius

Radius $r=\frac{d}{2}$, given $d = x$, so $r=\frac{x}{2}$.

Step2: Calculate the base - area

The formula for the area of the base of a cylinder (a circle) is $A=\pi r^{2}$. Substituting $r = \frac{x}{2}$, we get $A=\pi(\frac{x}{2})^{2}=\frac{1}{4}\pi x^{2}$.

Step3: Calculate the height

The volume formula of a cylinder is $V = A\times h=\pi r^{2}h$. Given $V=\pi x^{3}$ and $r=\frac{x}{2}$, then $\pi x^{3}=\pi(\frac{x}{2})^{2}h$. Solving for $h$:
\[

$$\begin{align*} \pi x^{3}&=\pi\times\frac{x^{2}}{4}\times h\\ h&=\frac{\pi x^{3}}{\frac{\pi x^{2}}{4}}\\ h& = 4x \end{align*}$$

\]

Answer:

The area of the cylinder's base is $\frac{1}{4}\pi x^{2}$ square units, The height of the cylinder is $4x$ units.