Given the problem, let's analyze the statements and determine which ones are true regarding the cylinder with a base diameter of [tex]\( x \)[/tex] units and a volume of [tex]\( \pi x^3 \)[/tex] cubic units.
Step-by-Step Analysis:
1. Determine the Radius of the Cylinder:
- The diameter of the base of the cylinder is [tex]\( x \)[/tex] units.
- The radius is half of the diameter.
[tex]\[
\text{Radius} = \frac{x}{2}
\][/tex]
2. Calculate the Area of the Cylinder's Base:
- The area of the base, which is a circle, uses the formula:
[tex]\[
\text{Base Area} = \pi \left(\frac{x}{2}\right)^2 = \pi \left(\frac{x^2}{4}\right) = \frac{1}{4} \pi x^2
\][/tex]
3. Determine the Height of the Cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \text{Base Area} \times \text{Height}
\][/tex]
- We are given that the volume is [tex]\( \pi x^3 \)[/tex].
- Using the base area from step 2:
[tex]\[
\pi x^3 = \left(\frac{1}{4} \pi x^2\right) \times \text{Height}
\][/tex]
- Solving for the height:
[tex]\[
\text{Height} = \frac{\pi x^3}{\frac{1}{4} \pi x^2} = \frac{\pi x^3}{\pi \frac{x^2}{4}} = \frac{4x^3}{x^2} = 4x
\][/tex]
4. Analyze the Statements:
- Statement 1: The radius of the cylinder is [tex]\(2x\)[/tex] units.
- We found the radius is [tex]\(\frac{x}{2}\)[/tex], not [tex]\(2x\)[/tex]. This statement is false.
- Statement 2: The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2\)[/tex] square units.
- We calculated the base area as [tex]\(\frac{1}{4} \pi x^2\)[/tex]. This statement is true.
- Statement 3: The area of the cylinder's base is [tex]\(\frac{1}{2} \pi x^2\)[/tex] square units.
- We calculated the base area as [tex]\(\frac{1}{4} \pi x^2\)[/tex], not [tex]\(\frac{1}{2} \pi x^2\)[/tex]. This statement is false.
- Statement 4: The height of the cylinder is [tex]\(2x\)[/tex] units.
- We determined the height to be [tex]\(4x\)[/tex], not [tex]\(2x\)[/tex]. This statement is false.
- Statement 5: The height of the cylinder is [tex]\(4x\)[/tex] units.
- We calculated the height as [tex]\(4x\)[/tex]. This statement is true.
Conclusion:
The two statements about the cylinder that are true are:
- "The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2\)[/tex] square units."
- "The height of the cylinder is [tex]\(4x\)[/tex] units."
