Answer :
To determine which statements about the cylinder are true, we need to analyze the properties of the cylinder given:
1. The volume of the cylinder is [tex]\( \pi x^3 \)[/tex] cubic units.
2. The base diameter of the cylinder is [tex]\( x \)[/tex] units.
### Step-by-Step Solution:
Step 1: Determine the 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]
Step 2: Verify the base area of the cylinder:
- The base area [tex]\( A \)[/tex] of a cylinder is given by [tex]\( \pi r^2 \)[/tex].
- Substituting the radius [tex]\( r = \frac{x}{2} \)[/tex]:
[tex]\[ A = \pi \left( \frac{x}{2} \right)^2 = \pi \left( \frac{x^2}{4} \right) = \frac{1}{4} \pi x^2 \][/tex]
- Therefore, the statement "The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units" is true.
Step 3: Verify the height of the cylinder [tex]\( h \)[/tex]:
- 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 base area [tex]\( \frac{1}{4} \pi x^2 \)[/tex]:
[tex]\[ \pi r^2 h = \pi x^3 \][/tex]
[tex]\[ \pi \left( \frac{x^2}{4} \right) h = \pi x^3 \][/tex]
[tex]\[ \frac{\pi x^2}{4} h = \pi x^3 \][/tex]
[tex]\[ \frac{x^2}{4} h = x^3 \][/tex]
[tex]\[ h = \frac{4x^3}{x^2} = 4x \][/tex]
- Therefore, the statement "The height of the cylinder is [tex]\( 4x \)[/tex] units" is true.
Given the information and the calculations:
1) The radius of the cylinder is [tex]$2x$[/tex] units. (False) - As calculated [tex]\( r = \frac{x}{2} \)[/tex].
2) The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units. (True)
3) The area of the cylinder's base is [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units. (False)
4) The height of the cylinder is [tex]\( 2x \)[/tex] units. (False) - The height was calculated to be [tex]\( 4x \)[/tex].
5) The height of the cylinder is [tex]\( 4x \)[/tex] units. (True)
The true 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.
1. The volume of the cylinder is [tex]\( \pi x^3 \)[/tex] cubic units.
2. The base diameter of the cylinder is [tex]\( x \)[/tex] units.
### Step-by-Step Solution:
Step 1: Determine the 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]
Step 2: Verify the base area of the cylinder:
- The base area [tex]\( A \)[/tex] of a cylinder is given by [tex]\( \pi r^2 \)[/tex].
- Substituting the radius [tex]\( r = \frac{x}{2} \)[/tex]:
[tex]\[ A = \pi \left( \frac{x}{2} \right)^2 = \pi \left( \frac{x^2}{4} \right) = \frac{1}{4} \pi x^2 \][/tex]
- Therefore, the statement "The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units" is true.
Step 3: Verify the height of the cylinder [tex]\( h \)[/tex]:
- 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 base area [tex]\( \frac{1}{4} \pi x^2 \)[/tex]:
[tex]\[ \pi r^2 h = \pi x^3 \][/tex]
[tex]\[ \pi \left( \frac{x^2}{4} \right) h = \pi x^3 \][/tex]
[tex]\[ \frac{\pi x^2}{4} h = \pi x^3 \][/tex]
[tex]\[ \frac{x^2}{4} h = x^3 \][/tex]
[tex]\[ h = \frac{4x^3}{x^2} = 4x \][/tex]
- Therefore, the statement "The height of the cylinder is [tex]\( 4x \)[/tex] units" is true.
Given the information and the calculations:
1) The radius of the cylinder is [tex]$2x$[/tex] units. (False) - As calculated [tex]\( r = \frac{x}{2} \)[/tex].
2) The area of the cylinder's base is [tex]\( \frac{1}{4} \pi x^2 \)[/tex] square units. (True)
3) The area of the cylinder's base is [tex]\( \frac{1}{2} \pi x^2 \)[/tex] square units. (False)
4) The height of the cylinder is [tex]\( 2x \)[/tex] units. (False) - The height was calculated to be [tex]\( 4x \)[/tex].
5) The height of the cylinder is [tex]\( 4x \)[/tex] units. (True)
The true 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.
