Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.