A bucket made up of a metal sheet is in the form of a frustum of cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the bucket if the cost metal sheet used is ₹ 15 per `100 cm^(2)`. [ Use `pi=3.14`]

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the given values
- Height of the frustum (h) = 16 cm
- Radius of the lower circular end (r) = 8 cm
- Radius of the upper circular end (R) = 20 cm
- Cost of metal sheet = ₹15 per 100 cm²

### Step 2: Calculate the slant height (l) of the frustum
The formula for the slant height \( l \) of a frustum of a cone is given by:
\[ l = \sqrt{(R - r)^2 + h^2} \]
Substituting the values:
\[ l = \sqrt{(20 - 8)^2 + 16^2} = \sqrt{(12)^2 + (16)^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm} \]

### Step 3: Calculate the total surface area (TSA) of the frustum
The formula for the total surface area of a frustum of a cone is:
\[ \text{TSA} = \pi (R + r) l + \pi r^2 \]
Since the bucket is open at the top, we only need the lateral surface area and the area of the lower circular base:
\[ \text{TSA} = \pi (R + r) l \]
Substituting the values:
\[ \text{TSA} = \pi (20 + 8) \cdot 20 \]
Using \( \pi = 3.14 \):
\[ \text{TSA} = 3.14 \cdot 28 \cdot 20 = 3.14 \cdot 560 = 1758.4 \text{ cm}^2 \]

### Step 4: Calculate the cost of the metal sheet used
First, we find the cost per cm²:
\[ \text{Cost per cm}^2 = \frac{15}{100} = 0.15 \text{ ₹} \]
Now, we calculate the total cost for the total surface area:
\[ \text{Total Cost} = \text{TSA} \cdot \text{Cost per cm}^2 = 1758.4 \cdot 0.15 \]
Calculating this gives:
\[ \text{Total Cost} = 263.76 \text{ ₹} \]

### Final Answer
The cost of the bucket is approximately ₹263.76.

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