Side of the square base of a pyramid is 8 cm. The pyramid is cut into parts by a plane parallel to its base such that heights of upper and lower parts are in the ratio 1:3 respectively. What is the ratio of volumes of lower and upper part respectively ?

Options

A. a)`27:1`
B. b)`63:1`
C. c)`64:1`
D. d)`26:1`

Answer

To find the ratio of the volumes of the lower and upper parts of the pyramid, we can follow these steps:

### Step 1: Understand the dimensions of the pyramid
The pyramid has a square base with a side length of 8 cm. The total height of the pyramid is divided into two parts, with the heights of the upper and lower parts in the ratio of 1:3.

### Step 2: Determine the total height of the pyramid
Since the heights of the upper and lower parts are in the ratio 1:3, we can denote the height of the upper part as \( h_1 \) and the height of the lower part as \( h_2 \). Thus, we can express:
- \( h_1 = x \)
- \( h_2 = 3x \)

The total height \( H \) of the pyramid is:
\[ H = h_1 + h_2 = x + 3x = 4x \]

### Step 3: Calculate the height of each part
From the above, we can see that:
- The height of the upper part \( h_1 = x \)
- The height of the lower part \( h_2 = 3x \)

### Step 4: Use the similarity of the pyramids
When the pyramid is cut by a plane parallel to the base, the upper part is similar to the original pyramid. The ratio of the heights of the upper part to the original pyramid is:
\[ \text{Ratio of heights} = \frac{h_1}{H} = \frac{x}{4x} = \frac{1}{4} \]

### Step 5: Calculate the volumes of the pyramids
The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

For the original pyramid:
- Base Area = \( 8 \times 8 = 64 \, \text{cm}^2 \)
- Height = \( 4x \)

Thus, the volume \( V_{original} \) is:
\[ V_{original} = \frac{1}{3} \times 64 \times 4x = \frac{256x}{3} \]

For the upper pyramid:
- Base Area = \( \left(\frac{1}{4} \times 8\right)^2 = 4^2 = 16 \, \text{cm}^2 \)
- Height = \( x \)

Thus, the volume \( V_{upper} \) is:
\[ V_{upper} = \frac{1}{3} \times 16 \times x = \frac{16x}{3} \]

### Step 6: Calculate the volume of the lower part
The volume of the lower part can be calculated by subtracting the volume of the upper part from the volume of the original pyramid:
\[ V_{lower} = V_{original} - V_{upper} = \frac{256x}{3} - \frac{16x}{3} = \frac{240x}{3} = 80x \]

### Step 7: Find the ratio of the volumes
Now, we can find the ratio of the volumes of the lower part to the upper part:
\[ \text{Ratio} = \frac{V_{lower}}{V_{upper}} = \frac{80x}{\frac{16x}{3}} = 80x \times \frac{3}{16x} = \frac{240}{16} = 15 \]

Thus, the ratio of the volumes of the lower and upper parts is:
\[ \text{Ratio} = 15:1 \]