A tent is in the shape of a right circular cylinder up to a height of 3 m and conical above it. The total height of the tent is 13.5 m and the radius of its base is 14 m. Find the cost of cloth required to make the tent at the rate of ₹ 80 per square metre.[Take `pi = 22/7]`

Answer

To find the cost of cloth required to make the tent, we need to calculate the curved surface area of the tent, which consists of a cylindrical part and a conical part. Here’s a step-by-step solution:

### Step 1: Identify the dimensions of the tent
- The height of the cylindrical part (h₁) = 3 m
- The total height of the tent (H) = 13.5 m
- The radius of the base (r) = 14 m

### Step 2: Calculate the height of the conical part
The height of the conical part (h₂) can be calculated as:
\[ h₂ = H - h₁ = 13.5 \, \text{m} - 3 \, \text{m} = 10.5 \, \text{m} \]

### Step 3: Calculate the slant height of the cone
To find the slant height (l) of the cone, we use the Pythagorean theorem:
\[ l = \sqrt{r^2 + h₂^2} \]
Substituting the values:
\[ l = \sqrt{(14 \, \text{m})^2 + (10.5 \, \text{m})^2} \]
\[ l = \sqrt{196 + 110.25} \]
\[ l = \sqrt{306.25} \]
\[ l = 17.5 \, \text{m} \]

### Step 4: Calculate the curved surface area of the cylinder
The curved surface area (CSA) of the cylindrical part is given by:
\[ \text{CSA}_{\text{cylinder}} = 2 \pi r h₁ \]
Substituting the values:
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 14 \, \text{m} \times 3 \, \text{m} \]
\[ \text{CSA}_{\text{cylinder}} = 2 \times \frac{22}{7} \times 42 \]
\[ \text{CSA}_{\text{cylinder}} = \frac{1848}{7} \, \text{m}^2 \]
\[ \text{CSA}_{\text{cylinder}} = 264 \, \text{m}^2 \]

### Step 5: Calculate the curved surface area of the cone
The curved surface area of the conical part is given by:
\[ \text{CSA}_{\text{cone}} = \pi r l \]
Substituting the values:
\[ \text{CSA}_{\text{cone}} = \frac{22}{7} \times 14 \, \text{m} \times 17.5 \, \text{m} \]
\[ \text{CSA}_{\text{cone}} = \frac{22}{7} \times 245 \]
\[ \text{CSA}_{\text{cone}} = \frac{5390}{7} \, \text{m}^2 \]
\[ \text{CSA}_{\text{cone}} = 770 \, \text{m}^2 \]

### Step 6: Calculate the total curved surface area of the tent
The total curved surface area (CSA_total) of the tent is the sum of the curved surface areas of the cylinder and the cone:
\[ \text{CSA}_{\text{total}} = \text{CSA}_{\text{cylinder}} + \text{CSA}_{\text{cone}} \]
\[ \text{CSA}_{\text{total}} = 264 \, \text{m}^2 + 770 \, \text{m}^2 \]
\[ \text{CSA}_{\text{total}} = 1034 \, \text{m}^2 \]

### Step 7: Calculate the cost of cloth required
The cost of cloth required is given by:
\[ \text{Cost} = \text{CSA}_{\text{total}} \times \text{rate} \]
Substituting the values:
\[ \text{Cost} = 1034 \, \text{m}^2 \times 80 \, \text{₹/m}^2 \]
\[ \text{Cost} = 82720 \, \text{₹} \]

### Final Answer
The cost of cloth required to make the tent is ₹ 82,720.

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