A container opened at the top and made up of a meta! 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 ? 50 per litre. Also find the cost of meta! sheet used to make the container, if it costs Rs. 10 per `100 cm^2` ? ( Take ` pi ` = 3.14).

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Calculate the Volume of the Frustum of the Cone
The formula for the volume \( V \) of a frustum of a cone is given by:

\[ V = \frac{1}{3} \pi h (R_1^2 + R_2^2 + R_1 R_2) \]

Where:
- \( h \) = height of the frustum = 16 cm
- \( R_1 \) = radius of the upper base = 20 cm
- \( R_2 \) = radius of the lower base = 8 cm
- \( \pi \) = 3.14

Substituting the values into the formula:

\[ V = \frac{1}{3} \times 3.14 \times 16 \times (20^2 + 8^2 + 20 \times 8) \]

Calculating \( R_1^2, R_2^2, \) and \( R_1 R_2 \):

\[ 20^2 = 400, \quad 8^2 = 64, \quad 20 \times 8 = 160 \]

Now, substituting these values back into the volume formula:

\[ V = \frac{1}{3} \times 3.14 \times 16 \times (400 + 64 + 160) \]
\[ = \frac{1}{3} \times 3.14 \times 16 \times 624 \]
\[ = \frac{1}{3} \times 3.14 \times 9984 \]
\[ = \frac{31353.76}{3} \approx 10451.25 \, \text{cm}^3 \]

### Step 2: Convert Volume to Liters
Since \( 1000 \, \text{cm}^3 = 1 \, \text{liter} \):

\[ \text{Volume in liters} = \frac{10451.25}{1000} \approx 10.45125 \, \text{liters} \]

### Step 3: Calculate the Cost of Milk
The cost of milk is given as Rs. 50 per liter. Therefore, the total cost of milk is:

\[ \text{Cost of milk} = 10.45125 \times 50 \approx 522.56 \, \text{Rs} \]

### Step 4: Calculate the Surface Area of the Frustum
The formula for the surface area \( A \) of a frustum of a cone (excluding the top) is:

\[ A = \pi (R_1 + R_2) l + \pi R_2^2 \]

Where \( l \) is the slant height, calculated using:

\[ l = \sqrt{h^2 + (R_1 - R_2)^2} \]

Calculating \( l \):

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

Now substituting into the surface area formula:

\[ A = \pi (20 + 8) \times 20 + \pi \times 8^2 \]
\[ = \pi \times 28 \times 20 + \pi \times 64 \]
\[ = 560\pi + 64\pi = 624\pi \]
\[ = 624 \times 3.14 \approx 1960.56 \, \text{cm}^2 \]

### Step 5: Calculate the Cost of the Metal Sheet
The cost of the metal sheet is Rs. 10 per 100 cm². Therefore, the total cost is:

\[ \text{Cost of metal sheet} = \frac{1960.56}{100} \times 10 \approx 196.056 \, \text{Rs} \]

### Final Answers
- Cost of milk: Rs. 522.56 (approximately Rs. 523)
- Cost of metal sheet: Rs. 196.06 (approximately Rs. 196)

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