A tent is made in the form of a frustum of a cone surmounted by anoter cone. The diametere of the base top of the frustum are 20m and 6m respectvely, and the height is 24m. If the height of the tent is 28 m and the radius of the conical part is equal to the radius of the top of the frustum, fidn the quantity of can vas required. [ Take `pi = 22/7`]

Explanation

To find the quantity of canvas required for the tent, we need to calculate the curved surface area of the frustum of the cone and the curved surface area of the cone on top of it. Here’s how we can do this step by step:

### Step 1: Identify the dimensions
- **Diameter of the base of the frustum (D1)** = 20 m, hence radius (R1) = D1/2 = 10 m
- **Diameter of the top of the frustum (D2)** = 6 m, hence radius (R2) = D2/2 = 3 m
- **Height of the frustum (h1)** = 24 m
- **Height of the cone (h2)** = Total height of the tent - height of the frustum = 28 m - 24 m = 4 m

### Step 2: Calculate the slant height of the cone (h2)
- The radius of the cone (R2) = 3 m
- The height of the cone (h2) = 4 m
- Using the Pythagorean theorem, the slant height (l2) of the cone can be calculated as:
\[ l2 = \sqrt{(R2)^2 + (h2)^2} = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ m} \]

### Step 3: Calculate the slant height of the frustum (l1)
- The radius of the base of the frustum (R1) = 10 m
- The radius of the top of the frustum (R2) = 3 m
- The height of the frustum (h1) = 24 m
- The slant height (l1) of the frustum can be calculated using the Pythagorean theorem:
\[ l1 = \sqrt{(R1 - R2)^2 + (h1)^2} = \sqrt{(10 - 3)^2 + (24)^2} = \sqrt{(7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ m} \]

### Step 4: Calculate the curved surface area of the frustum
The formula for the curved surface area (CSA) of a frustum of a cone is:
\[ \text{CSA}_{\text{frustum}} = \pi (R1 + R2) l1 \]
Substituting the values:
\[ \text{CSA}_{\text{frustum}} = \frac{22}{7} \times (10 + 3) \times 25 = \frac{22}{7} \times 13 \times 25 = \frac{22 \times 325}{7} = \frac{7150}{7} \approx 1021.43 \text{ m}^2 \]

### Step 5: Calculate the curved surface area of the cone
The formula for the curved surface area of a cone is:
\[ \text{CSA}_{\text{cone}} = \pi R2 l2 \]
Substituting the values:
\[ \text{CSA}_{\text{cone}} = \frac{22}{7} \times 3 \times 5 = \frac{22 \times 15}{7} = \frac{330}{7} \approx 47.14 \text{ m}^2 \]

### Step 6: Calculate the total canvas required
The total canvas required is the sum of the curved surface areas of the frustum and the cone:
\[ \text{Total CSA} = \text{CSA}_{\text{frustum}} + \text{CSA}_{\text{cone}} \approx 1021.43 + 47.14 \approx 1068.57 \text{ m}^2 \]

### Final Answer
The quantity of canvas required is approximately **1068.57 m²**.