To find the lateral surface area of a cylinder, we can use the formula:
\[ \text{Lateral Surface Area} = 2 \pi r h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.
From the dimensions provided:
- The height \( h \) of the cylinder is 3 cm.
- The width of the rectangle (which corresponds to the diameter of the cylinder) is 9 cm. Therefore, the radius \( r \) is half of the diameter:
\[ r = \frac{9 , \text{cm}}{2} = 4.5 , \text{cm} \]
Now, substituting the values into the formula:
\[ \text{Lateral Surface Area} = 2 \pi (4.5 , \text{cm}) (3 , \text{cm}) \]
Calculating this step-by-step:
- Calculate the product of the radius and height:
\[ 4.5 , \text{cm} \times 3 , \text{cm} = 13.5 , \text{cm}^2 \]
- Multiply by 2:
\[ 2 \times 13.5 , \text{cm}^2 = 27 , \text{cm}^2 \]
- Now multiply by \( \pi \):
\[ \text{Lateral Surface Area} = 27 \pi , \text{cm}^2 \]
Using \( \pi \approx 3.14 \):
\[ \text{Lateral Surface Area} \approx 27 \times 3.14 \approx 84.78 , \text{cm}^2 \]
None of the provided options perfectly match this result, but the closest approximation for \( 27 \pi , \text{cm}^2 \) is about 84.78 cm². If you have other rounding methods or precise values of \( \pi \), check for the more accurate number.
Given that the options provided do not match this calculation correctly, the calculations align with the correct understanding of lateral area but may differ based on interpretations or values for π in other contexts. However, none of the options A, B, C, or D correctly match the computed lateral surface area.