A circus tent is in the form of a right circular cylinder and right circular cone above it . The diameter and the height of the cylindrical part of the tent are 126 m and 5 m respectively.The totla height of the tent is 21 m. Find the total cost of the tent if the canvas used costsRs 12 per square metre.

Answer

To solve the problem, we need to find the total surface area of the circus tent, which consists of a cylindrical part and a conical part, and then calculate the cost based on the area and the cost per square meter.

### Step 1: Find the radius and height of the cylindrical part.
- The diameter of the cylindrical part is given as 126 m.
- Therefore, the radius \( r \) of the cylindrical part is:
\[ r = \frac{\text{diameter}}{2} = \frac{126}{2} = 63 \, \text{m} \]
- The height \( h_c \) of the cylindrical part is given as 5 m.

### Step 2: Find the height of the conical part.
- The total height of the tent is given as 21 m.
- The height \( h_{cone} \) of the conical part can be calculated as:
\[ h_{cone} = \text{Total height} - \text{Height of cylindrical part} = 21 - 5 = 16 \, \text{m} \]

### Step 3: Calculate the surface area of the cylindrical part.
- The formula for the lateral surface area \( A_c \) of a cylinder is:
\[ A_c = 2\pi rh_c \]
- Substituting the values:
\[ A_c = 2 \times \pi \times 63 \times 5 \]
\[ A_c = 630\pi \, \text{m}^2 \]

### Step 4: Calculate the slant height of the conical part.
- The slant height \( l \) of the cone can be found using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h_{cone}^2} = \sqrt{63^2 + 16^2} \]
\[ l = \sqrt{3969 + 256} = \sqrt{4225} = 65 \, \text{m} \]

### Step 5: Calculate the surface area of the conical part.
- The formula for the lateral surface area \( A_{cone} \) of a cone is:
\[ A_{cone} = \pi r l \]
- Substituting the values:
\[ A_{cone} = \pi \times 63 \times 65 \]
\[ A_{cone} = 4095\pi \, \text{m}^2 \]

### Step 6: Calculate the total surface area of the tent.
- The total surface area \( A_{total} \) is the sum of the surface areas of the cylindrical and conical parts:
\[ A_{total} = A_c + A_{cone} = 630\pi + 4095\pi = 4725\pi \, \text{m}^2 \]

### Step 7: Calculate the cost of the canvas.
- The cost per square meter is given as Rs 12.
- Therefore, the total cost \( C \) is:
\[ C = A_{total} \times \text{cost per square meter} = 4725\pi \times 12 \]
- Approximating \( \pi \) as 3.14:
\[ C \approx 4725 \times 3.14 \times 12 \]
\[ C \approx 4725 \times 37.68 \approx 177,078 \, \text{Rs} \]

### Final Answer:
The total cost of the tent is approximately Rs 177,078.

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