A bucket is in the form of a frustum of a cone and hold 28.490 litres of water. The radii of the top and bottom are 28 cm and 21 cm respectively. Find the height of the bucket.

Answer

To find the height of the bucket, which is in the form of a frustum of a cone, we will use the formula for the volume of a frustum. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Convert Volume from Liters to Cubic Centimeters**:
- Given volume = 28.490 liters.
- Since 1 liter = 1000 cm³, we convert:
\[ \text{Volume} = 28.490 \times 1000 = 28490 \, \text{cm}^3 \]

2. **Identify the Radii of the Frustum**:
- The radius of the top (r1) = 28 cm.
- The radius of the bottom (r2) = 21 cm.

3. **Use the Formula for the Volume of a Frustum**:
- The formula for the volume \( V \) of a frustum of a cone is:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]
- Here, we will use \( \pi \approx \frac{22}{7} \).

4. **Substitute the Known Values into the Formula**:
- Set \( V = 28490 \, \text{cm}^3 \), \( r_1 = 28 \, \text{cm} \), \( r_2 = 21 \, \text{cm} \):
\[ 28490 = \frac{1}{3} \times \frac{22}{7} \times h \times (28^2 + 21^2 + 28 \times 21) \]

5. **Calculate \( r_1^2 + r_2^2 + r_1 r_2 \)**:
- Calculate \( 28^2 = 784 \),
- Calculate \( 21^2 = 441 \),
- Calculate \( 28 \times 21 = 588 \).
- Now sum these values:
\[ 784 + 441 + 588 = 1813 \]

6. **Substitute Back into the Volume Equation**:
- Now substitute back into the equation:
\[ 28490 = \frac{1}{3} \times \frac{22}{7} \times h \times 1813 \]

7. **Simplify the Equation**:
- Multiply both sides by 3 to eliminate the fraction:
\[ 3 \times 28490 = \frac{22}{7} \times h \times 1813 \]
- Calculate \( 3 \times 28490 = 85470 \).

8. **Multiply \( \frac{22}{7} \) and \( 1813 \)**:
- Calculate \( \frac{22 \times 1813}{7} \):
\[ \frac{22 \times 1813}{7} = \frac{39986}{7} \approx 5712.29 \]

9. **Set Up the Final Equation**:
- Now we have:
\[ 85470 = 5712.29 \times h \]

10. **Solve for \( h \)**:
- Divide both sides by 5712.29:
\[ h = \frac{85470}{5712.29} \approx 14.95 \, \text{cm} \]
- Rounding gives us approximately \( h \approx 15 \, \text{cm} \).

### Final Answer:
The height of the bucket is approximately **15 cm**.