The diameters of the two circular ends of the bucket are 44 cm and 24 cm. The height of the bucket is 35cm. Find the volume of the bucket.

Answer

To find the volume of the bucket, which is in the shape of a frustum of a cone, we will follow these steps:

### Step 1: Identify the given dimensions
- The diameter of the upper circular end (D1) = 44 cm
- The diameter of the lower circular end (D2) = 24 cm
- The height of the bucket (h) = 35 cm

### Step 2: Calculate the radii
- The radius of the upper circular end (R1) = D1 / 2 = 44 cm / 2 = 22 cm
- The radius of the lower circular end (R2) = D2 / 2 = 24 cm / 2 = 12 cm

### Step 3: Use the formula for the volume of a frustum
The formula for the volume (V) of a frustum of a cone is:
\[ V = \frac{1}{3} \pi h (R1^2 + R2^2 + R1 \cdot R2) \]

### Step 4: Substitute the values into the formula
Using \(\pi \approx \frac{22}{7}\):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 35 \times (22^2 + 12^2 + 22 \cdot 12) \]

### Step 5: Calculate \(R1^2\), \(R2^2\), and \(R1 \cdot R2\)
- \(R1^2 = 22^2 = 484\)
- \(R2^2 = 12^2 = 144\)
- \(R1 \cdot R2 = 22 \cdot 12 = 264\)

### Step 6: Add these values together
\[ R1^2 + R2^2 + R1 \cdot R2 = 484 + 144 + 264 = 892 \]

### Step 7: Substitute back into the volume formula
\[ V = \frac{1}{3} \times \frac{22}{7} \times 35 \times 892 \]

### Step 8: Simplify the expression
First, calculate \(\frac{1}{3} \times 35 = \frac{35}{3}\):
\[ V = \frac{22}{7} \times \frac{35}{3} \times 892 \]
Now, calculate \(\frac{22 \times 35 \times 892}{21}\).

### Step 9: Calculate the final volume
Calculating \(22 \times 35 = 770\):
\[ V = \frac{770 \times 892}{21} \]
Calculating \(770 \times 892 = 686840\):
\[ V = \frac{686840}{21} \approx 32692.38 \text{ cm}^3 \]

### Final Answer
The volume of the bucket is approximately \(32692.38 \text{ cm}^3\).

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