A solid consists of circular cylinder with exact fitting right circular cone placed on the top. The height of the cone is h. If total volume of the solid is three times the volume of the cone, then the height of the circular cylinder is

Options

A. `2h`
B. `(2h)/(3) `
C. `4h`
D. `(3h)/(2) `

Answer

To solve the problem step by step, we need to find the height of the circular cylinder given that the total volume of the solid (cylinder + cone) is three times the volume of the cone.

### Step-by-Step Solution:

1. **Understand the Volumes**:
- The volume \( V_c \) of a right circular cone is given by the formula:
\[ V_c = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

2. **Volume of the Cylinder**:
- Let \( H \) be the height of the circular cylinder. The volume \( V_{cy} \) of the circular cylinder is given by:
\[ V_{cy} = \pi r^2 H \]

3. **Total Volume of the Solid**:
- The total volume \( V_{total} \) of the solid (cylinder + cone) can be expressed as:
\[ V_{total} = V_c + V_{cy} = \frac{1}{3} \pi r^2 h + \pi r^2 H \]

4. **Given Condition**:
- According to the problem, the total volume of the solid is three times the volume of the cone:
\[ V_{total} = 3 V_c \]
- Substituting the volume of the cone:
\[ \frac{1}{3} \pi r^2 h + \pi r^2 H = 3 \left( \frac{1}{3} \pi r^2 h \right) \]

5. **Simplifying the Equation**:
- The right side simplifies to:
\[ \pi r^2 h \]
- Therefore, we have:
\[ \frac{1}{3} \pi r^2 h + \pi r^2 H = \pi r^2 h \]
- We can cancel \( \pi r^2 \) from all terms (assuming \( r \neq 0 \)):
\[ \frac{1}{3} h + H = h \]

6. **Isolate \( H \)**:
- Rearranging the equation gives:
\[ H = h - \frac{1}{3} h \]
- This simplifies to:
\[ H = \frac{3h}{3} - \frac{1h}{3} = \frac{2h}{3} \]

7. **Final Result**:
- The height of the circular cylinder \( H \) is:
\[ H = \frac{2h}{3} \]