A bucket in the shape of the frustum of a cone has its top and bottom radii as 20 cm and 10 cm, respectively. The depth of the bucket is 24 cm The capacity of the bucket is (Take `pi = (22)/7`)

Options

A. `8800 cm^3`
B. `13200 cm^2`
C. `17000 cm^2`
D. `17600 cm^3`

Answer

To find the capacity of the bucket in the shape of a frustum of a cone, we will use the formula for the volume of a frustum:

\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]

where:
- \( V \) is the volume,
- \( h \) is the height (depth) of the frustum,
- \( r_1 \) is the radius of the top base,
- \( r_2 \) is the radius of the bottom base.

### Step 1: Identify the given values
- Top radius (\( r_1 \)) = 20 cm
- Bottom radius (\( r_2 \)) = 10 cm
- Height (\( h \)) = 24 cm
- \( \pi \) = \( \frac{22}{7} \)

### Step 2: Substitute the values into the volume formula
\[ V = \frac{1}{3} \times \frac{22}{7} \times 24 \times (20^2 + 10^2 + 20 \times 10) \]

### Step 3: Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 \times r_2 \)
- \( r_1^2 = 20^2 = 400 \)
- \( r_2^2 = 10^2 = 100 \)
- \( r_1 \times r_2 = 20 \times 10 = 200 \)

### Step 4: Sum the squares and product
\[ r_1^2 + r_2^2 + r_1 \times r_2 = 400 + 100 + 200 = 700 \]

### Step 5: Substitute back into the volume formula
\[ V = \frac{1}{3} \times \frac{22}{7} \times 24 \times 700 \]

### Step 6: Simplify the expression
First, calculate \( \frac{1}{3} \times 24 = 8 \):
\[ V = \frac{22}{7} \times 8 \times 700 \]

### Step 7: Calculate \( 8 \times 700 \)
\[ 8 \times 700 = 5600 \]

### Step 8: Substitute back into the volume formula
\[ V = \frac{22}{7} \times 5600 \]

### Step 9: Calculate \( \frac{22 \times 5600}{7} \)
First, calculate \( 22 \times 5600 = 123200 \):
\[ V = \frac{123200}{7} \]

### Step 10: Perform the division
\[ V = 17600 \text{ cm}^3 \]

### Final Answer
The capacity of the bucket is \( 17600 \text{ cm}^3 \).

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