An open metal bucket is in the shape of a frustum of cone of height 21 cm with radii of its lower and upper ends are 10 cm and 20 cm respectively. Find the cost of milk which can completely fill the bucket at the rate of रु 40 per litre.

Answer

To solve the problem step by step, we will find the volume of the frustum of the cone and then calculate the cost of the milk needed to fill the bucket.

### Step 1: Identify the dimensions of the frustum of the cone
- Height (h) = 21 cm
- Radius of the lower end (r) = 10 cm
- Radius of the upper end (R) = 20 cm

### Step 2: Use the formula for the volume of a frustum of a cone
The volume \( V \) of a frustum of a cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi h (R^2 + r^2 + Rr) \]
Where:
- \( \pi \) is approximately \( \frac{22}{7} \) (or you can use 3.14)
- \( h \) is the height
- \( R \) is the radius of the upper base
- \( r \) is the radius of the lower base

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 21 \times (20^2 + 10^2 + 20 \times 10) \]
Calculating \( 20^2 + 10^2 + 20 \times 10 \):
\[ 20^2 = 400, \quad 10^2 = 100, \quad 20 \times 10 = 200 \]
So,
\[ 20^2 + 10^2 + 20 \times 10 = 400 + 100 + 200 = 700 \]
Now substituting back into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 21 \times 700 \]

### Step 4: Simplify the expression
Calculating:
\[ V = \frac{1}{3} \times \frac{22 \times 21 \times 700}{7} \]
First, simplify \( \frac{21}{7} = 3 \):
\[ V = \frac{1}{3} \times 22 \times 3 \times 700 \]
Now, \( 3 \) cancels out:
\[ V = 22 \times 700 = 15400 \text{ cm}^3 \]

### Step 5: Convert volume to liters
Since 1 liter = 1000 cm³, we convert the volume:
\[ \text{Volume in liters} = \frac{15400}{1000} = 15.4 \text{ liters} \]

### Step 6: Calculate the cost of the milk
The cost of milk is given as ₹40 per liter. Therefore, the total cost \( C \) is:
\[ C = 15.4 \times 40 \]
Calculating:
\[ C = 616 \text{ rupees} \]

### Final Answer
The cost of milk which can completely fill the bucket is ₹616.

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