Answer :
To answer the question about the cylinder with a base diameter of [tex]\( x \)[/tex] units and a volume of [tex]\( \pi x^3 \)[/tex] cubic units, we need to determine the radius, the area of the base, and the height of the cylinder. Let's break down the problem step by step:
1. Radius of the Cylinder:
- The diameter of the base is given as [tex]\( x \)[/tex] units.
- So, the radius [tex]\( r \)[/tex] is half of the diameter: [tex]\( r = \frac{x}{2} \)[/tex].
2. Volume of the Cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder can be expressed as [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- Given [tex]\( V = \pi x^3 \)[/tex], we substitute the radius into the volume formula: [tex]\( \pi \left(\frac{x}{2}\right)^2 h = \pi x^3 \)[/tex].
- Simplify the equation to find [tex]\( h \)[/tex]:
[tex]\[ \pi \frac{x^2}{4} h = \pi x^3 \][/tex]
[tex]\[ \frac{x^2}{4} h = x^3 \][/tex]
[tex]\[ h = \frac{4x^3}{x^2} = 4x \][/tex]
3. Area of the Cylinder's Base:
- The area [tex]\( A \)[/tex] of the base of the cylinder is given by the formula [tex]\( A = \pi r^2 \)[/tex].
- Using 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]
With these calculations, we now evaluate each of the given statements:
1. "The radius of the cylinder is [tex]\( 2x \)[/tex] units."
- This statement is False. The radius is [tex]\(\frac{x}{2}\)[/tex] units, not [tex]\(2x\)[/tex] units.
2. "The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2 \)[/tex] square units."
- This statement is True. We calculated the base area to be [tex]\( \frac{1}{4} \pi x^2 \)[/tex].
3. "The area of the cylinder's base is [tex]\(\frac{1}{2} \pi x^2 \)[/tex] square units."
- This statement is False. The area is [tex]\( \frac{1}{4} \pi x^2 \)[/tex], not [tex]\( \frac{1}{2} \pi x^2 \)[/tex].
4. "The height of the cylinder is [tex]\( 2 x \)[/tex] units."
- This statement is False. The height is [tex]\( 4x \)[/tex] units, not [tex]\( 2x \)[/tex] units.
5. "The height of the cylinder is [tex]\( 4 x \)[/tex] units."
- This statement is True. We calculated the height to be [tex]\( 4x \)[/tex].
Conclusively, the true statements about the cylinder 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]\( 4 x \)[/tex] units.
1. Radius of the Cylinder:
- The diameter of the base is given as [tex]\( x \)[/tex] units.
- So, the radius [tex]\( r \)[/tex] is half of the diameter: [tex]\( r = \frac{x}{2} \)[/tex].
2. Volume of the Cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder can be expressed as [tex]\( V = \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- Given [tex]\( V = \pi x^3 \)[/tex], we substitute the radius into the volume formula: [tex]\( \pi \left(\frac{x}{2}\right)^2 h = \pi x^3 \)[/tex].
- Simplify the equation to find [tex]\( h \)[/tex]:
[tex]\[ \pi \frac{x^2}{4} h = \pi x^3 \][/tex]
[tex]\[ \frac{x^2}{4} h = x^3 \][/tex]
[tex]\[ h = \frac{4x^3}{x^2} = 4x \][/tex]
3. Area of the Cylinder's Base:
- The area [tex]\( A \)[/tex] of the base of the cylinder is given by the formula [tex]\( A = \pi r^2 \)[/tex].
- Using 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]
With these calculations, we now evaluate each of the given statements:
1. "The radius of the cylinder is [tex]\( 2x \)[/tex] units."
- This statement is False. The radius is [tex]\(\frac{x}{2}\)[/tex] units, not [tex]\(2x\)[/tex] units.
2. "The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2 \)[/tex] square units."
- This statement is True. We calculated the base area to be [tex]\( \frac{1}{4} \pi x^2 \)[/tex].
3. "The area of the cylinder's base is [tex]\(\frac{1}{2} \pi x^2 \)[/tex] square units."
- This statement is False. The area is [tex]\( \frac{1}{4} \pi x^2 \)[/tex], not [tex]\( \frac{1}{2} \pi x^2 \)[/tex].
4. "The height of the cylinder is [tex]\( 2 x \)[/tex] units."
- This statement is False. The height is [tex]\( 4x \)[/tex] units, not [tex]\( 2x \)[/tex] units.
5. "The height of the cylinder is [tex]\( 4 x \)[/tex] units."
- This statement is True. We calculated the height to be [tex]\( 4x \)[/tex].
Conclusively, the true statements about the cylinder 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]\( 4 x \)[/tex] units.
