To solve the problem, we need to find two things: the volume of the frustum of the cone (which will give us the amount of milk it can hold) and the surface area of the frustum (which will help us calculate the cost of the metal sheet used to make the container).
### 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:
- \( h \) is the height of the frustum,
- \( r_1 \) is the radius of the lower base,
- \( r_2 \) is the radius of the upper base.
Given:
- \( h = 16 \, \text{cm} \)
- \( r_1 = 8 \, \text{cm} \)
- \( r_2 = 20 \, \text{cm} \)
Substituting the values into the formula:
\[
V = \frac{1}{3} \times 3.14 \times 16 \times (8^2 + 20^2 + 8 \times 20)
\]
Calculating the squares and product:
\[
8^2 = 64, \quad 20^2 = 400, \quad 8 \times 20 = 160
\]
Now, substituting these values:
\[
V = \frac{1}{3} \times 3.14 \times 16 \times (64 + 400 + 160)
\]
\[
= \frac{1}{3} \times 3.14 \times 16 \times 624
\]
Calculating the sum:
\[
64 + 400 + 160 = 624
\]
Now calculating the volume:
\[
V = \frac{1}{3} \times 3.14 \times 16 \times 624
\]
\[
= \frac{1}{3} \times 3.14 \times 9984
\]
\[
= \frac{1}{3} \times 31376.16
\]
\[
= 10458.72 \, \text{cm}^3
\]
### Step 2: Convert Volume to Liters
Since \( 1 \, \text{liter} = 1000 \, \text{cm}^3 \):
\[
\text{Volume in liters} = \frac{10458.72}{1000} = 10.45872 \, \text{liters}
\]
### Step 3: Calculate the Cost of Milk
The cost of milk is given as Rs 20 per liter. Therefore, the total cost for the milk is:
\[
\text{Cost of milk} = 10.45872 \times 20 = 209.1744 \approx 209.17 \, \text{Rs}
\]
### Step 4: Calculate the Curved Surface Area of the Frustum
The formula for the curved surface area \( A \) of a frustum of a cone is given by:
\[
A = \pi (r_1 + r_2) l
\]
Where \( l \) is the slant height, which can be calculated using the Pythagorean theorem:
\[
l = \sqrt{(r_2 - r_1)^2 + h^2}
\]
Calculating \( l \):
\[
l = \sqrt{(20 - 8)^2 + 16^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \, \text{cm}
\]
Now substituting into the surface area formula:
\[
A = 3.14 \times (8 + 20) \times 20
\]
\[
= 3.14 \times 28 \times 20
\]
\[
= 3.14 \times 560
\]
\[
= 1758.4 \, \text{cm}^2
\]
### Step 5: Calculate the Cost of the Metal Sheet
The cost of the metal sheet is Rs 8 per 100 cm². Therefore, the total cost for the metal sheet is:
\[
\text{Cost of metal sheet} = \frac{1758.4}{100} \times 8 = 140.672 \approx 140.67 \, \text{Rs}
\]
### Final Answers
- Cost of milk: Rs 209.17
- Cost of metal sheet: Rs 140.67
