A tent is of the shape of a right circular cylinder upto height of 3 metres and then becomes a right circular cone with a maximum height of 13.5 metres above the ground. Calculate the cost of painting the inner surface of the tent at Rs 4 per sq. metre, if the radius of the base is 14 metres.

Answer

To solve the problem step by step, we will calculate the inner surface area of the tent, which consists of a cylindrical part and a conical part, and then determine the cost of painting that surface area.

### Step 1: Identify the dimensions of the tent
- The height of the cylindrical part (h_cylinder) = 3 meters
- The height of the conical part (h_cone) = Total height - Height of cylindrical part = 13.5 meters - 3 meters = 10.5 meters
- The radius of the base (r) = 14 meters

### Step 2: Calculate the slant height of the conical part
To find the slant height (l) of the conical part, we use the Pythagorean theorem:
\[ l = \sqrt{r^2 + h_{cone}^2} \]
Substituting the values:
\[ l = \sqrt{14^2 + 10.5^2} \]
Calculating:
\[ l = \sqrt{196 + 110.25} = \sqrt{306.25} = 17.5 \text{ meters} \]

### Step 3: Calculate the curved surface area of the conical part
The formula for the curved surface area (CSA) of a cone is:
\[ CSA_{cone} = \pi r l \]
Substituting the values:
\[ CSA_{cone} = \frac{22}{7} \times 14 \times 17.5 \]
Calculating:
\[ CSA_{cone} = \frac{22}{7} \times 14 \times 17.5 = 770 \text{ square meters} \]

### Step 4: Calculate the curved surface area of the cylindrical part
The formula for the curved surface area of a cylinder is:
\[ CSA_{cylinder} = 2 \pi r h \]
Substituting the values:
\[ CSA_{cylinder} = 2 \times \frac{22}{7} \times 14 \times 3 \]
Calculating:
\[ CSA_{cylinder} = 2 \times \frac{22}{7} \times 14 \times 3 = 264 \text{ square meters} \]

### Step 5: Calculate the total curved surface area of the tent
Now, we add the curved surface areas of the cone and the cylinder:
\[ Total \, CSA = CSA_{cone} + CSA_{cylinder} \]
Substituting the values:
\[ Total \, CSA = 770 + 264 = 1034 \text{ square meters} \]

### Step 6: Calculate the cost of painting
The cost of painting is given as Rs 4 per square meter. Thus, the total cost of painting is:
\[ Cost = Total \, CSA \times \text{Cost per square meter} \]
Substituting the values:
\[ Cost = 1034 \times 4 = 4136 \text{ rupees} \]

### Final Answer
The total cost of painting the inner surface of the tent is **Rs 4136**.

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