a cylinder with a base diameter of x units has a volume of $pi x^{3}$ cubic units. which statements about…

a cylinder with a base diameter of x units has a volume of $pi x^{3}$ 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 $\frac{1}{4}pi x^{2}$ square units. the area of the cylinders base is $\frac{1}{2}pi x^{2}$ square units. the height of the cylinder is 2x units. the height of the cylinder is 4x units.

a cylinder with a base diameter of x units has a volume of $pi x^{3}$ 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 $\frac{1}{4}pi x^{2}$ square units. the area of the cylinders base is $\frac{1}{2}pi x^{2}$ square units. the height of the cylinder is 2x units. the height of the cylinder is 4x units.

Answer

Explanation:

Step1: Find the radius

The diameter of the base is $x$ units. The radius $r$ of a circle is half of the diameter, so $r=\frac{x}{2}$ units.

Step2: Calculate the base - area

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

Step3: Use the volume formula to find the height

The volume formula of a cylinder is $V=Ah$, where $V$ is the volume, $A$ is the base - area, and $h$ is the height. Given $V = \pi x^{3}$ and $A=\frac{1}{4}\pi x^{2}$. Then $h=\frac{V}{A}=\frac{\pi x^{3}}{\frac{1}{4}\pi x^{2}}$. Simplifying the right - hand side, $\frac{\pi x^{3}}{\frac{1}{4}\pi x^{2}} = 4x$ units.

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.