The radii of the circular ends of a frustum of a cone are 20 cm and 13 cm and its height is 12 cm. What is the capacity (in litres) of the frustum (correct to one decimal place)? (Take `pi=(22)/(7)`)

Options

A. 10.4
B. 11.2
C. 10.8
D. 11.4

Answer

To find the capacity of the 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,
- \( \pi \) is approximately \( \frac{22}{7} \),
- \( h \) is the height of the frustum,
- \( r_1 \) is the radius of the larger circular end,
- \( r_2 \) is the radius of the smaller circular end.

### Step 1: Identify the values
From the problem, we have:
- \( r_1 = 20 \) cm
- \( r_2 = 13 \) cm
- \( h = 12 \) cm
- \( \pi = \frac{22}{7} \)

### Step 2: Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 r_2 \)
- \( r_1^2 = 20^2 = 400 \)
- \( r_2^2 = 13^2 = 169 \)
- \( r_1 r_2 = 20 \times 13 = 260 \)

### Step 3: Substitute the values into the volume formula
Now we substitute these values into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 12 \times (400 + 169 + 260) \]

### Step 4: Calculate the sum inside the parentheses
Calculate \( 400 + 169 + 260 \):

\[ 400 + 169 + 260 = 829 \]

### Step 5: Substitute back into the volume formula
Now substitute back into the formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 12 \times 829 \]

### Step 6: Calculate the volume
First, calculate \( \frac{1}{3} \times 12 = 4 \):

\[ V = 4 \times \frac{22}{7} \times 829 \]

Now calculate \( 4 \times \frac{22}{7} = \frac{88}{7} \):

\[ V = \frac{88}{7} \times 829 \]

Now perform the multiplication:

\[ V = \frac{88 \times 829}{7} = \frac{72952}{7} \approx 10422.57 \text{ cm}^3 \]

### Step 7: Convert cm³ to liters
Since \( 1 \text{ cm}^3 = 0.001 \text{ liters} \):

\[ V \approx 10422.57 \times 0.001 \approx 10.42257 \text{ liters} \]

### Step 8: Round to one decimal place
Rounding \( 10.42257 \) to one decimal place gives us:

\[ V \approx 10.4 \text{ liters} \]

### Final Answer
The capacity of the frustum is approximately **10.4 liters**.

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