An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for fold and for stitching. Give your answer to the nearest `m ^(2).`

Answer

To find the quantity of canvas required to make the exhibition tent, which is in the form of a cylinder surmounted by a cone, we will follow these steps:

### Step 1: Identify the Given Values
- Total height of the tent (H) = 85 m
- Height of the cylindrical part (h_cylinder) = 50 m
- Diameter of the base (D) = 168 m

### Step 2: Calculate the Height of the Cone
The height of the conical part (h_cone) can be calculated as:
\[ h_{cone} = H - h_{cylinder} = 85 \, \text{m} - 50 \, \text{m} = 35 \, \text{m} \]

### Step 3: Calculate the Radius of the Base
The radius (r) of the base can be calculated from the diameter:
\[ r = \frac{D}{2} = \frac{168 \, \text{m}}{2} = 84 \, \text{m} \]

### Step 4: Calculate the Slant Height of the Cone
The slant height (l) of the cone can be calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h_{cone}^2} \]
Substituting the values:
\[ l = \sqrt{84^2 + 35^2} \]
Calculating the squares:
\[ 84^2 = 7056 \quad \text{and} \quad 35^2 = 1225 \]
Adding them:
\[ l = \sqrt{7056 + 1225} = \sqrt{8281} = 91 \, \text{m} \]

### Step 5: Calculate the Curved Surface Area of the Cone
The curved surface area (CSA) of the cone is given by:
\[ CSA_{cone} = \pi r l \]
Substituting the values:
\[ CSA_{cone} = \frac{22}{7} \times 84 \times 91 \]

### Step 6: Calculate the Curved Surface Area of the Cylinder
The curved surface area of the cylinder is given by:
\[ CSA_{cylinder} = 2 \pi r h_{cylinder} \]
Substituting the values:
\[ CSA_{cylinder} = 2 \times \frac{22}{7} \times 84 \times 50 \]

### Step 7: Calculate the Total Curved Surface Area
The total curved surface area (TCSA) required for the canvas is:
\[ TCSA = CSA_{cone} + CSA_{cylinder} \]

### Step 8: Calculate the Total Quantity of Canvas Required
Now, we will calculate the total quantity of canvas required, allowing for 20% extra for folds and stitching:
\[ \text{Total Canvas} = TCSA + 0.2 \times TCSA = 1.2 \times TCSA \]

### Step 9: Final Calculation
Now we will compute the values:
1. Calculate \( CSA_{cone} \):
\[ CSA_{cone} = \frac{22}{7} \times 84 \times 91 = 22 \times 12 \times 91 = 24024 \, \text{m}^2 \]
2. Calculate \( CSA_{cylinder} \):
\[ CSA_{cylinder} = 2 \times \frac{22}{7} \times 84 \times 50 = \frac{44}{7} \times 84 \times 50 = 26400 \, \text{m}^2 \]
3. Calculate \( TCSA \):
\[ TCSA = 24024 + 26400 = 50424 \, \text{m}^2 \]
4. Calculate the total canvas required:
\[ \text{Total Canvas} = 1.2 \times 50424 = 60508.8 \, \text{m}^2 \]
Rounding to the nearest square meter:
\[ \text{Total Canvas} \approx 60509 \, \text{m}^2 \]

### Final Answer
The quantity of canvas required to make the tent is approximately **60509 m²**.

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