The ratio of the heights of a right circular cone and a right circular cylinder is 8:9 and the ratio of the radii of their bases is 3:4. If the volume of the cylinder is 132 cubic cm , then the volume (in cubic cm) of the cone is:

Options

A. 44
B. 22
C. 66
D. 33

Answer

To solve the problem, we need to find the volume of a right circular cone given the volume of a right circular cylinder and the ratios of their heights and radii.

### Step-by-Step Solution:

1. **Identify the Given Ratios:**
- The ratio of the heights of the cone (h_c) to the cylinder (h_cyl) is 8:9.
- The ratio of the radii of the cone (r_c) to the cylinder (r_cyl) is 3:4.

2. **Express Heights and Radii in Terms of Variables:**
- Let the height of the cone be \( h_c = 8x \) and the height of the cylinder be \( h_cyl = 9x \).
- Let the radius of the cone be \( r_c = 3y \) and the radius of the cylinder be \( r_cyl = 4y \).

3. **Volume Formulas:**
- The volume of the cone (V_cone) is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r_c^2 h_c \]
- The volume of the cylinder (V_cyl) is given by the formula:
\[ V_{cyl} = \pi r_cyl^2 h_cyl \]

4. **Substituting the Values:**
- Substitute the expressions for the radius and height of the cone into the volume formula:
\[ V_{cone} = \frac{1}{3} \pi (3y)^2 (8x) = \frac{1}{3} \pi (9y^2)(8x) = 24\pi xy^2 \]
- Substitute the expressions for the radius and height of the cylinder into the volume formula:
\[ V_{cyl} = \pi (4y)^2 (9x) = \pi (16y^2)(9x) = 144\pi xy^2 \]

5. **Setting Up the Ratio of Volumes:**
- The ratio of the volumes of the cone to the cylinder can be expressed as:
\[ \frac{V_{cone}}{V_{cyl}} = \frac{24\pi xy^2}{144\pi xy^2} = \frac{24}{144} = \frac{1}{6} \]

6. **Using the Given Volume of the Cylinder:**
- We know the volume of the cylinder \( V_{cyl} = 132 \, \text{cm}^3 \).
- From the ratio, we can find the volume of the cone:
\[ V_{cone} = \frac{1}{6} \times 132 = 22 \, \text{cm}^3 \]

### Final Answer:
The volume of the cone is \( 22 \, \text{cm}^3 \).