Answer :
To solve this problem, we need to derive the radius, base area, and height of the cylinder from the given information.
1. Radius of the Cylinder:
- The diameter of the cylinder's base is given as [tex]\( x \)[/tex] units.
- The radius is half the diameter. Thus, the radius [tex]\(\text{r}\)[/tex] is:
[tex]\[ \text{radius} = \frac{x}{2} \][/tex]
2. Base Area of the Cylinder:
- The base area of the cylinder is calculated using the formula for the area of a circle, which is [tex]\(\pi \text{r}^2\)[/tex].
- Substituting the radius we found:
[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. Height of the Cylinder:
- The volume of the cylinder is given by the formula:
[tex]\[ \text{volume} = \text{base area} \times \text{height} \][/tex]
- Given that the volume is [tex]\(\pi x^3\)[/tex]:
[tex]\[ \pi x^3 = \frac{1}{4} \pi x^2 \times \text{height} \][/tex]
- Solving for the height:
[tex]\[ \text{height} = \frac{\pi x^3}{\frac{1}{4} \pi x^2} = \frac{4 \pi x^3}{\pi x^2} = 4 x \][/tex]
Next, let's evaluate the given options:
1. The radius of the cylinder is [tex]\(2 x\)[/tex] units.
- This statement is false; the radius is [tex]\(\frac{x}{2}\)[/tex].
2. The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2\)[/tex] square units.
- This statement is true; the calculated base area is [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 correct base area is [tex]\(\frac{1}{4} \pi x^2\)[/tex].
4. The height of the cylinder is [tex]\(2 x\)[/tex] units.
- This statement is false; the calculated height is [tex]\(4 x\)[/tex].
5. The height of the cylinder is [tex]\(4 x\)[/tex] units.
- This statement is true; the calculated height is [tex]\(4 x\)[/tex].
Therefore, the two true statements about the cylinder 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.
1. Radius of the Cylinder:
- The diameter of the cylinder's base is given as [tex]\( x \)[/tex] units.
- The radius is half the diameter. Thus, the radius [tex]\(\text{r}\)[/tex] is:
[tex]\[ \text{radius} = \frac{x}{2} \][/tex]
2. Base Area of the Cylinder:
- The base area of the cylinder is calculated using the formula for the area of a circle, which is [tex]\(\pi \text{r}^2\)[/tex].
- Substituting the radius we found:
[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. Height of the Cylinder:
- The volume of the cylinder is given by the formula:
[tex]\[ \text{volume} = \text{base area} \times \text{height} \][/tex]
- Given that the volume is [tex]\(\pi x^3\)[/tex]:
[tex]\[ \pi x^3 = \frac{1}{4} \pi x^2 \times \text{height} \][/tex]
- Solving for the height:
[tex]\[ \text{height} = \frac{\pi x^3}{\frac{1}{4} \pi x^2} = \frac{4 \pi x^3}{\pi x^2} = 4 x \][/tex]
Next, let's evaluate the given options:
1. The radius of the cylinder is [tex]\(2 x\)[/tex] units.
- This statement is false; the radius is [tex]\(\frac{x}{2}\)[/tex].
2. The area of the cylinder's base is [tex]\(\frac{1}{4} \pi x^2\)[/tex] square units.
- This statement is true; the calculated base area is [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 correct base area is [tex]\(\frac{1}{4} \pi x^2\)[/tex].
4. The height of the cylinder is [tex]\(2 x\)[/tex] units.
- This statement is false; the calculated height is [tex]\(4 x\)[/tex].
5. The height of the cylinder is [tex]\(4 x\)[/tex] units.
- This statement is true; the calculated height is [tex]\(4 x\)[/tex].
Therefore, the two true statements about the cylinder 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.
