A container in the shape of a frustum of a cone having diameters of its two circular faces as 35 cm and 30 cm and vertical height 14 cm, is completely filled with oil. If each `cm^(3)` of oil has mass 1.2 g then find the cost of oil in the container if it costs ₹40 per kg.

Explanation

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

### Step 1: Identify the dimensions of the frustum of the cone
- The diameter of the larger circular face (D) = 35 cm
- The diameter of the smaller circular face (d) = 30 cm
- The vertical height (h) = 14 cm

### Step 2: Calculate the radii of the circular faces
- Radius of the larger circular face (R) = D/2 = 35 cm / 2 = 17.5 cm
- Radius of the smaller circular face (r) = d/2 = 30 cm / 2 = 15 cm

### Step 3: Use the formula for the volume of a frustum of a cone
The formula for the volume (V) of a frustum of a cone is given by:
\[ V = \frac{1}{3} \pi h (R^2 + r^2 + Rr) \]
Substituting the values we have:
- \( \pi \approx \frac{22}{7} \)
- \( h = 14 \) cm
- \( R = 17.5 \) cm
- \( r = 15 \) cm

### Step 4: Substitute the values into the volume formula
\[ V = \frac{1}{3} \times \frac{22}{7} \times 14 \times (17.5^2 + 15^2 + 17.5 \times 15) \]

### Step 5: Calculate \( R^2 \), \( r^2 \), and \( Rr \)
- \( R^2 = (17.5)^2 = 306.25 \)
- \( r^2 = (15)^2 = 225 \)
- \( Rr = 17.5 \times 15 = 262.5 \)

### Step 6: Calculate the total inside the brackets
\[ R^2 + r^2 + Rr = 306.25 + 225 + 262.5 = 793.75 \]

### Step 7: Substitute back into the volume formula
\[ V = \frac{1}{3} \times \frac{22}{7} \times 14 \times 793.75 \]

### Step 8: Simplify the expression
First, calculate \( \frac{22}{7} \times 14 \):
\[ \frac{22 \times 14}{7} = 44 \]
Now substitute:
\[ V = \frac{1}{3} \times 44 \times 793.75 \]
Calculating \( 44 \times 793.75 \):
\[ 44 \times 793.75 = 34945 \]
Now divide by 3:
\[ V = \frac{34945}{3} \approx 11648.33 \text{ cm}^3 \]

### Step 9: Calculate the mass of the oil
Given that each \( cm^3 \) of oil has a mass of 1.2 grams:
\[ \text{Mass} = 11648.33 \times 1.2 \approx 13977.996 \text{ grams} \]

### Step 10: Convert mass to kilograms
\[ \text{Mass in kg} = \frac{13977.996}{1000} \approx 13.978 \text{ kg} \]

### Step 11: Calculate the cost of the oil
Given the cost of oil is ₹40 per kg:
\[ \text{Cost} = 13.978 \times 40 \approx 559.12 \]

### Final Answer
The cost of the oil in the container is approximately ₹559.12.

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