The height of a solid right circular cylinder is 6 metres and three times the sum of the area of its two end faces is twice the area of its curved surface. The radius of its base (in metre) is

Options

A. 4
B. 2
C. 8
D. 10

Answer

To solve the problem, we need to find the radius of a solid right circular cylinder given its height and a relationship between the areas of its end faces and its curved surface.

### Step-by-Step Solution:

1. **Identify the given values:**
- Height (h) of the cylinder = 6 meters.
- Let the radius of the base be r meters.

2. **Calculate the area of the two end faces:**
- The area of one end face (circle) = πr².
- Therefore, the area of two end faces = 2 × πr² = 2πr².

3. **Calculate the curved surface area (CSA) of the cylinder:**
- The formula for the curved surface area of a cylinder = 2πrh.
- Substituting the height, we get CSA = 2πr(6) = 12πr.

4. **Set up the equation based on the problem statement:**
- According to the problem, three times the sum of the area of its two end faces is equal to twice the area of its curved surface.
- This can be expressed as:
\[ 3 \times (2πr²) = 2 \times (12πr) \]

5. **Simplify the equation:**
- Left side: \( 3 \times 2πr² = 6πr² \).
- Right side: \( 2 \times 12πr = 24πr \).
- Therefore, we have:
\[ 6πr² = 24πr \]

6. **Divide both sides by 6π (assuming π ≠ 0):**
\[ r² = 4r \]

7. **Rearranging the equation:**
\[ r² - 4r = 0 \]

8. **Factor the equation:**
\[ r(r - 4) = 0 \]

9. **Solve for r:**
- This gives us two solutions: \( r = 0 \) or \( r = 4 \).
- Since the radius cannot be zero, we have:
\[ r = 4 \text{ meters} \]

### Final Answer:
The radius of the base of the cylinder is **4 meters**.