Let's analyze the problem step by step.
We are given:
- The base diameter of the cylinder is [tex]\( x \)[/tex] units.
- The volume of the cylinder is [tex]\( \pi x^3 \)[/tex] cubic units.
From this information, we can derive certain properties of the cylinder.
### Step 1: Determine the radius of the cylinder.
The diameter [tex]\( d \)[/tex] of the cylinder is given as [tex]\( x \)[/tex] units.
The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{x}{2} \][/tex]
### Step 2: Calculate the area of the cylinder's base.
The area [tex]\( A \)[/tex] of a circle (which forms the base of the cylinder) is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
Substituting [tex]\( r = \frac{x}{2} \)[/tex]:
[tex]\[ A = \pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4} = \frac{1}{4} \pi x^2 \][/tex]
Therefore, the area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units, which confirms one of the given options.
### Step 3: Relate volume to height.
The volume [tex]\( V \)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
We know the volume [tex]\( V = \pi x^3 \)[/tex] and the radius [tex]\( r = \frac{x}{2} \)[/tex]. Substitute these into the volume formula:
[tex]\[ \pi x^3 = \pi \left(\frac{x}{2}\right)^2 h \][/tex]
[tex]\[ \pi x^3 = \pi \left(\frac{x^2}{4}\right) h \][/tex]
[tex]\[ \pi x^3 = \frac{\pi x^2}{4} h \][/tex]
Now, solve for [tex]\( h \)[/tex]:
[tex]\[ x^3 = \frac{x^2}{4} h \][/tex]
[tex]\[ 4 x^3 = x^2 h \][/tex]
[tex]\[ h = 4 x \][/tex]
So, the height of the cylinder is [tex]\( 4 x \)[/tex] units, confirming another given option.
### Conclusion:
Based on the analysis:
- The radius of the cylinder is [tex]\( \frac{x}{2} \)[/tex] units, not [tex]\( 2 x \)[/tex] units.
- The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units. (True)
- The area of the cylinder's base is not [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units.
- The height of the cylinder is not [tex]\( 2 x \)[/tex] units.
- The height of the cylinder is [tex]\( 4 x \)[/tex] units. (True)
Therefore, the true statements are:
1. The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units.
2. The height of the cylinder is [tex]\( 4 x \)[/tex] units.
