The radius of the ends of a bucket of height 24 cm are 15 cm and 5 cm. Find its capacity .Also find the surface area of the buket .

Answer

To solve the problem, we need to find the capacity (volume) and surface area of a bucket that has a height of 24 cm, a top radius of 15 cm, and a bottom radius of 5 cm. The bucket is in the shape of a frustum of a cone.

### Step 1: Write down the given values
- Height of the bucket (h) = 24 cm
- Radius of the top (r1) = 15 cm
- Radius of the bottom (r2) = 5 cm

### Step 2: Calculate the volume of the bucket
The formula for the volume \( V \) of a frustum of a cone is given by:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]

Substituting the known values into the formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 24 \times (15^2 + 5^2 + 15 \times 5) \]

### Step 3: Calculate \( r_1^2 \), \( r_2^2 \), and \( r_1 \times r_2 \)
- \( r_1^2 = 15^2 = 225 \)
- \( r_2^2 = 5^2 = 25 \)
- \( r_1 \times r_2 = 15 \times 5 = 75 \)

Now, add these values:
\[ r_1^2 + r_2^2 + r_1 \times r_2 = 225 + 25 + 75 = 325 \]

### Step 4: Substitute back into the volume formula
Now substituting back into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 24 \times 325 \]

### Step 5: Calculate the volume
Calculating the volume step-by-step:
1. Calculate \( \frac{1}{3} \times 24 = 8 \)
2. Now calculate \( 8 \times \frac{22}{7} \times 325 \)

Calculating \( 8 \times 22 = 176 \):
\[ V = \frac{176 \times 325}{7} \]

Calculating \( 176 \times 325 = 57200 \):
\[ V = \frac{57200}{7} \approx 8171.43 \text{ cm}^3 \]

### Step 6: Calculate the surface area of the bucket
The formula for the surface area \( A \) of a frustum of a cone is given by:
\[ A = \pi (r_1 + r_2) l + \pi r_2^2 \]
where \( l \) is the slant height.

### Step 7: Calculate the slant height \( l \)
Using the Pythagorean theorem:
\[ l = \sqrt{h^2 + (r_1 - r_2)^2} \]
Substituting the values:
\[ l = \sqrt{24^2 + (15 - 5)^2} = \sqrt{576 + 100} = \sqrt{676} = 26 \text{ cm} \]

### Step 8: Substitute into the surface area formula
Now substituting into the surface area formula:
\[ A = \frac{22}{7} (15 + 5) \times 26 + \frac{22}{7} \times 5^2 \]
\[ A = \frac{22}{7} \times 20 \times 26 + \frac{22}{7} \times 25 \]

### Step 9: Calculate the surface area
Calculating:
1. \( 20 \times 26 = 520 \)
2. \( A = \frac{22}{7} \times 520 + \frac{22}{7} \times 25 \)
3. \( A = \frac{22}{7} (520 + 25) = \frac{22}{7} \times 545 \)

Calculating \( 22 \times 545 = 11990 \):
\[ A = \frac{11990}{7} \approx 1714.29 \text{ cm}^2 \]

### Final Answers
- The capacity of the bucket is approximately \( 8171.43 \text{ cm}^3 \).
- The surface area of the bucket is approximately \( 1714.29 \text{ cm}^2 \).