A right prism has a square base with side of base 4 cm and the height of prism is 9 cm. The prism is cut in three parts of equal heights by two planes parallel to its base. What is the ratio of the volume of the top, middle and the bottom part respectively?

Options

A. 1 : 8 : 27
B. 1 : 1 : 1
C. 1 : 8 : 20
D. 1 : 7 : 20

Answer

To solve the problem step by step, we will calculate the volumes of the top, middle, and bottom parts of the prism after it has been cut into three equal height sections.

### Step-by-Step Solution:

1. **Identify the dimensions of the prism:**
- The base of the prism is a square with a side length of 4 cm.
- The height of the prism is 9 cm.

2. **Calculate the volume of the entire prism:**
- The formula for the volume \( V \) of a prism is given by:
\[ V = \text{Base Area} \times \text{Height} \]
- The area of the square base \( A \) is:
\[ A = \text{side}^2 = 4^2 = 16 \text{ cm}^2 \]
- Therefore, the volume of the prism is:
\[ V = 16 \text{ cm}^2 \times 9 \text{ cm} = 144 \text{ cm}^3 \]

3. **Determine the height of each part after cutting:**
- The prism is cut into three parts of equal height. Since the total height is 9 cm, the height of each part is:
\[ \text{Height of each part} = \frac{9 \text{ cm}}{3} = 3 \text{ cm} \]

4. **Calculate the volume of each part:**
- Since the height of each part is the same and the base area remains constant, the volume of each part can be calculated as follows:
\[ V_{\text{each part}} = \text{Base Area} \times \text{Height of each part} = 16 \text{ cm}^2 \times 3 \text{ cm} = 48 \text{ cm}^3 \]

5. **Determine the ratio of the volumes of the top, middle, and bottom parts:**
- Since all three parts have the same volume:
\[ V_{\text{top}} = V_{\text{middle}} = V_{\text{bottom}} = 48 \text{ cm}^3 \]
- Therefore, the ratio of the volumes of the top, middle, and bottom parts is:
\[ \text{Ratio} = 48 : 48 : 48 = 1 : 1 : 1 \]

### Final Answer:
The ratio of the volume of the top, middle, and bottom parts respectively is **1 : 1 : 1**.