The diameters of the two circular ends of a bucket are 44 cm and 24 cm. The height of the bucket is 35 cm. The capacity of the bucket is (in litres)
32.7 litres
33.7 litres
34.7 litres
31.7 litres
Step 1: Identify the shape
The bucket is in the shape of a frustum of a cone (a cone with its top portion cut off).
Step 2: Write the formula for the volume of a frustum
\[
V = \dfrac{1}{3} \pi h (R^2 + Rr + r^2)
\]
where
- \(R\) = radius of the bigger circular end,
- \(r\) = radius of the smaller circular end,
- \(h\) = height of the frustum.
Step 3: Find the radii
Given diameters: 44 cm and 24 cm.
\(R = \dfrac{44}{2} = 22\,\text{cm}\),
\(r = \dfrac{24}{2} = 12\,\text{cm}\).
Step 4: Height
Height of the bucket: \(h = 35\,\text{cm}\).
Step 5: Substitute values in the formula
\[ V = \dfrac{1}{3} \pi (35) (22^2 + 22 \times 12 + 12^2) \]
First calculate inside the brackets:
\(22^2 = 484\),
\(22 \times 12 = 264\),
\(12^2 = 144\).
So, \(484 + 264 + 144 = 892\).
Step 6: Multiply
\[ V = \dfrac{1}{3} \pi (35)(892) \]
\(35 \times 892 = 31,220\).
\[ V = \dfrac{1}{3} \pi (31,220) \]
\(31,220 \div 3 = 10,406.67\).
\[ V \approx 3.1416 \times 10,406.67 = 32,689.1\,\text{cm}^3 \]
Step 7: Convert to litres
We know \(1000\,\text{cm}^3 = 1\,\text{L}\).
\[ V = \dfrac{32,689.1}{1000} = 32.7\,\text{L} \]
Final Answer: The capacity of the bucket is 32.7 litres (Option A).