Question

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.
\\(\square\\) the radius of the cylinder is 2x units.
\\(\square\\) the area of the cylinders base is $\frac{1}{4}\pi x^2$ square units.
\\(\square\\) the area of the cylinders base is $\frac{1}{2}\pi x^2$ square units.
\\(\square\\) the height of the cylinder is 2x units.
\\(\square\\) the height of the cylinder is 4x units.

Explanation:

Step1: Find the radius

The diameter of the base is \( x \) units, so the radius \( r=\frac{x}{2} \) units. So the first statement "The radius of the cylinder is \( 2x \) units" is false.

Step2: Calculate the area of the base

The area of a circle (base of the cylinder) is \( A = \pi r^{2} \). Substituting \( r=\frac{x}{2} \), we get \( A=\pi(\frac{x}{2})^{2}=\frac{1}{4}\pi x^{2} \) square units. So the second statement "The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units" is true, and the third statement is false.

Step3: Use the volume formula to find the height

The volume of a cylinder is \( V=\pi r^{2}h \). We know \( V = \pi x^{3} \) and \( r=\frac{x}{2} \), so \( \pi x^{3}=\pi(\frac{x}{2})^{2}h \). Simplifying, \( \pi x^{3}=\frac{1}{4}\pi x^{2}h \). Divide both sides by \( \frac{1}{4}\pi x^{2} \), we get \( h = 4x \) units? Wait, no, wait: Wait, \( \pi x^{3}=\pi(\frac{x}{2})^{2}h\Rightarrow x^{3}=\frac{x^{2}}{4}h\Rightarrow h = 4x \)? Wait, no, wait, let's do it again. \( \pi x^{3}=\pi(\frac{x}{2})^{2}h \). Divide both sides by \( \pi \): \( x^{3}=\frac{x^{2}}{4}h \). Then divide both sides by \( \frac{x^{2}}{4} \): \( h=\frac{x^{3}}{\frac{x^{2}}{4}}=4x \)? Wait, but let's check the options. Wait, the fourth option is "The height of the cylinder is \( 2x \) units", fifth is "The height of the cylinder is \( 4x \) units". Wait, maybe I made a mistake. Wait, volume \( V=\pi r^{2}h \), \( r = \frac{x}{2} \), so \( V=\pi(\frac{x}{2})^{2}h=\frac{\pi x^{2}}{4}h \). We know \( V=\pi x^{3} \), so \( \frac{\pi x^{2}}{4}h=\pi x^{3} \). Divide both sides by \( \pi x^{2} \): \( \frac{h}{4}=x \), so \( h = 4x \) units. Wait, but let's check the options again. Wait, the fourth option is "The height of the cylinder is \( 2x \) units" (false), fifth is "The height of the cylinder is \( 4x \) units" (true)? Wait, but earlier when we calculated the base area, the second statement is true. Wait, but the problem says to select two options. Wait, maybe I made a mistake in the height calculation. Wait, let's re - calculate the height. \( V=\pi r^{2}h \), \( V = \pi x^{3} \), \( r=\frac{x}{2} \), so:

\( \pi x^{3}=\pi(\frac{x}{2})^{2}h \)

\( x^{3}=\frac{x^{2}}{4}h \)

Multiply both sides by 4: \( 4x^{3}=x^{2}h \)

Divide both sides by \( x^{2} \) (assuming \( x eq0 \)): \( h = 4x \). Wait, but the fifth option is "The height of the cylinder is \( 4x \) units", and the second option is "The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units". Wait, but in the original problem, there is a typo? Wait, no, let's re - check. Wait, the radius is \( \frac{x}{2} \), so area of base is \( \pi(\frac{x}{2})^{2}=\frac{1}{4}\pi x^{2} \), that's correct. Then volume \( V=\pi r^{2}h=\frac{1}{4}\pi x^{2}h=\pi x^{3} \), so \( \frac{1}{4}h=x \), so \( h = 4x \). Wait, but the fourth option is "The height of the cylinder is \( 2x \) units" (false), fifth is "The height of the cylinder is \( 4x \) units" (true). And the second option "The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units" (true). Wait, but in the original problem, the third option is written as "The area of the cylinder's base is \( \frac{1}{2}\pi x^{2} \) square units" (false). So the correct statements are the second one (area of base is \( \frac{1}{4}\pi x^{2} \)) and the fifth one (height is \( 4x \))? Wait, no, wait, let's recalculate the height again. Wait, \( V=\pi r^{2}h \), \( V = \pi x^{3} \), \( r=\frac{x}{2} \), so:

\( \pi x^{3}=\pi(\frac{x}{2})^{2}h \)

\( x^{3}=\frac{x^{2}}{4}h \)

\( h=\frac{x^{3}}{\frac{x^{2}}{4}} = 4x \). Yes, so height is \( 4x \) units (fifth option is true). And the area of the base is \( \frac{1}{4}\pi x^{2} \) (second option is true). Wait, but in the original problem, the second option is "The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units" (correct), and the fifth option is "The height of the cylinder is \( 4x \) units" (correct). Wait, but maybe I made a mistake. Wait, no, let's check again. Radius \( r=\frac{x}{2} \), area of base \( A=\pi r^{2}=\pi(\frac{x}{2})^{2}=\frac{1}{4}\pi x^{2} \), correct. Volume \( V = A\times h=\frac{1}{4}\pi x^{2}\times h=\pi x^{3} \), so \( h=\frac{\pi x^{3}}{\frac{1}{4}\pi x^{2}} = 4x \), correct. So the two true statements are:

- The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units.

- The height of the cylinder is \( 4x \) units.

Wait, but in the original problem, the second option is "The area of the cylinder's base is \( \frac{1}{4}\pi x^{2} \) square units" (correct), and the fifth option is "The height of the cylinder is \( 4x \) units" (correct). Also, wait, maybe there is a mistake in my calculation. Wait, no, let's take \( x = 2 \) as an example. Diameter \( x = 2 \), so radius \( r = 1 \). Area of base \( A=\pi(1)^{2}=\pi \). According to the formula \( \frac{1}{4}\pi x^{2} \), when \( x = 2 \), \( \frac{1}{4}\pi(4)=\pi \), which matches. Volume \( V=\pi x^{3}=\pi(8) = 8\pi \). Using \( V = A\times h=\pi\times h \), so \( h = 8 \). And \( 4x=8 \) (since \( x = 2 \)), which matches. So the height is \( 4x \) units, and the area of the base is \( \frac{1}{4}\pi x^{2} \) square units.

Answer:

B. The area of the cylinder's base is \(\frac{1}{4}\pi x^{2}\) square units.

E. The height of the cylinder is \(4x\) units.

(Assuming the options are labeled as A: The radius of the cylinder is \(2x\) units, B: The area of the cylinder's base is \(\frac{1}{4}\pi x^{2}\) square units, C: The area of the cylinder's base is \(\frac{1}{2}\pi x^{2}\) square units, D: The height of the cylinder is \(2x\) units, E: The height of the cylinder is \(4x\) units)