A container, open from the top, made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container at the rate of Rs 15 per litre and the cost of metal sheet used, if it costs Rs 5 per 100 `cm^(2)`. (take`pi` = 22/7)

Answer

To solve the problem step by step, we will first calculate the volume of the frustum of the cone, then the cost of the milk that can fill the container, and finally the cost of the metal sheet used to make the container.

### Step 1: Calculate the slant height (L) of the frustum
The formula for the slant height \( L \) of a frustum of a cone is given by:
\[ L = \sqrt{h^2 + (R_2 - R_1)^2} \]
Where:
- \( h = 16 \) cm (height of the frustum)
- \( R_1 = 8 \) cm (radius of the lower end)
- \( R_2 = 20 \) cm (radius of the upper end)

Calculating:
\[ L = \sqrt{16^2 + (20 - 8)^2} = \sqrt{256 + 144} = \sqrt{400} = 20 \text{ cm} \]

### Step 2: Calculate the volume of the frustum
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 values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 16 \times (8^2 + 20^2 + 8 \times 20) \]
Calculating the squares and products:
\[ = \frac{1}{3} \times \frac{22}{7} \times 16 \times (64 + 400 + 160) \]
\[ = \frac{1}{3} \times \frac{22}{7} \times 16 \times 624 \]
Calculating further:
\[ = \frac{22 \times 16 \times 624}{3 \times 7} = \frac{22 \times 16 \times 624}{21} \]
Calculating \( 22 \times 16 = 352 \):
\[ = \frac{352 \times 624}{21} = \frac{219648}{21} \approx 10450.2857 \text{ cm}^3 \]

### Step 3: Convert volume to liters
Since \( 1 \text{ liter} = 1000 \text{ cm}^3 \):
\[ V \approx 10.45 \text{ liters} \]

### Step 4: Calculate the cost of milk
The cost of milk is given as Rs 15 per liter. Therefore, the cost for 10.45 liters is:
\[ \text{Cost of milk} = 10.45 \times 15 = 156.75 \text{ Rs} \]

### Step 5: Calculate the curved surface area (CSA) of the frustum
The formula for the curved surface area \( CSA \) of a frustum is:
\[ CSA = \pi (R_1 + R_2) L \]

Substituting the values:
\[ CSA = \frac{22}{7} \times (8 + 20) \times 20 \]
Calculating:
\[ = \frac{22}{7} \times 28 \times 20 = \frac{22 \times 560}{7} = \frac{12320}{7} \approx 1760 \text{ cm}^2 \]

### Step 6: Calculate the cost of the metal sheet
The cost of the metal sheet is Rs 5 per 100 cm². Therefore, the cost of the metal sheet is:
\[ \text{Cost of metal sheet} = \frac{5}{100} \times 1760 = 88 \text{ Rs} \]

### Final Summary:
1. Cost of milk = Rs 156.75
2. Cost of metal sheet = Rs 88.00