To find the area of the tin sheet required to make the oil funnel, we need to calculate the curved surface area of both the cylindrical portion and the frustum of the cone. Here’s how to do it step by step:
### Step 1: Identify the dimensions
- Length of the cylindrical portion (H_cylinder) = 10 cm
- Total height of the funnel (H_total) = 22 cm
- Diameter of the cylindrical portion = 8 cm
- Diameter of the top of the funnel = 18 cm
### Step 2: Calculate the radius of the cylindrical portion
The radius of the cylindrical portion (R_cylinder) is half of the diameter:
\[
R_{cylinder} = \frac{8 \, \text{cm}}{2} = 4 \, \text{cm}
\]
### Step 3: Calculate the radius of the frustum's top
The radius of the top of the funnel (R1) is half of the diameter:
\[
R_1 = \frac{18 \, \text{cm}}{2} = 9 \, \text{cm}
\]
### Step 4: Calculate the radius of the frustum's base
The radius of the base of the frustum (R2) is the same as the radius of the cylindrical portion:
\[
R_2 = R_{cylinder} = 4 \, \text{cm}
\]
### Step 5: Calculate the height of the frustum
The height of the frustum (H_frustum) can be found by subtracting the height of the cylindrical portion from the total height:
\[
H_{frustum} = H_{total} - H_{cylinder} = 22 \, \text{cm} - 10 \, \text{cm} = 12 \, \text{cm}
\]
### Step 6: Calculate the slant height of the frustum
Using the formula for the slant height (L) of a frustum:
\[
L = \sqrt{H_{frustum}^2 + (R_1 - R_2)^2}
\]
Substituting the values:
\[
L = \sqrt{12^2 + (9 - 4)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{cm}
\]
### Step 7: Calculate the curved surface area of the cylindrical portion
The formula for the curved surface area (CSA) of a cylinder is:
\[
CSA_{cylinder} = 2 \pi R_{cylinder} H_{cylinder}
\]
Substituting the values:
\[
CSA_{cylinder} = 2 \pi (4 \, \text{cm}) (10 \, \text{cm}) = 80 \pi \, \text{cm}^2
\]
### Step 8: Calculate the curved surface area of the frustum
The formula for the curved surface area of a frustum is:
\[
CSA_{frustum} = \pi (R_1 + R_2) L
\]
Substituting the values:
\[
CSA_{frustum} = \pi (9 \, \text{cm} + 4 \, \text{cm}) (13 \, \text{cm}) = \pi (13)(13) = 169 \pi \, \text{cm}^2
\]
### Step 9: Calculate the total area of the tin sheet required
The total area of the tin sheet required is the sum of the curved surface areas of the cylinder and the frustum:
\[
\text{Total Area} = CSA_{cylinder} + CSA_{frustum} = 80 \pi + 169 \pi = 249 \pi \, \text{cm}^2
\]
### Step 10: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[
\text{Total Area} = 249 \times \frac{22}{7} \, \text{cm}^2 = \frac{5478}{7} \, \text{cm}^2
\]
Calculating this gives:
\[
\text{Total Area} \approx 782.57 \, \text{cm}^2
\]
### Final Answer
The area of the tin sheet required to make the funnel is approximately \(782.57 \, \text{cm}^2\).
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