Let's analyze the information step by step:
1. Radius of the cylinder:
- The diameter of the cylinder is [tex]\( x \)[/tex] units.
- The radius, [tex]\( r \)[/tex], is half of the diameter, so [tex]\( r = \frac{x}{2} \)[/tex].
2. Volume of the cylinder:
- The volume of a cylinder is given by the formula [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- We are given that the volume is [tex]\( \pi x^3 \)[/tex] cubic units.
Therefore:
[tex]\[
\pi \left( \frac{x}{2} \right)^2 h = \pi x^3
\][/tex]
Simplifying inside the parentheses:
[tex]\[
\pi \frac{x^2}{4} h = \pi x^3
\][/tex]
Dividing both sides by [tex]\( \pi \)[/tex]:
[tex]\[
\frac{x^2}{4} h = x^3
\][/tex]
To isolate [tex]\( h \)[/tex], we multiply both sides by 4:
[tex]\[
x^2 h = 4 x^3
\][/tex]
Finally, dividing both sides by [tex]\( x^2 \)[/tex]:
[tex]\[
h = 4x
\][/tex]
3. Base area of the cylinder:
- The base area, [tex]\( A \)[/tex], of the cylinder is the area of the circle with radius [tex]\( r \)[/tex], which is given by [tex]\( A = \pi r^2 \)[/tex].
- Substituting the radius [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]
Now, let's verify each statement:
- Statement 1: The radius of the cylinder is [tex]\( 2x \)[/tex] units.
- This is incorrect because we found the radius to be [tex]\( \frac{x}{2} \)[/tex] units.
- Statement 2: The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units.
- This is correct as shown in the calculation above.
- Statement 3: The area of the cylinder's base is [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units.
- This is incorrect because the base area is [tex]\( \frac{1}{4} \pi x^2 \)[/tex].
- Statement 4: The height of the cylinder is [tex]\( 2x \)[/tex] units.
- This is incorrect because we found the height to be [tex]\( 4x \)[/tex] units.
- Statement 5: The height of the cylinder is [tex]\( 4x \)[/tex] units.
- This is correct as shown in the calculation above.
Therefore, the two correct statements 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.
