To find the lateral surface area of a cylinder, we use the formula:
\[ \text{Lateral Surface Area} = 2\pi rh \]
where \( r \) is the radius of the cylinder's base, and \( h \) is the height.
From your description, the height of the cylinder is 3 cm. However, the width of the rectangle (mentioned as 9 cm) is not the radius but rather the diameter of the cylinder. Therefore, to find the radius, we need to divide the diameter by 2:
\[ r = \frac{\text{Diameter}}{2} = \frac{9 , \text{cm}}{2} = 4.5 , \text{cm} \]
Now, substituting the values of \( r \) and \( h \) into the lateral surface area formula:
\[ \text{Lateral Surface Area} = 2\pi (4.5)(3) \]
Calculating this step-by-step:
- Calculate the product \( 2 \times 4.5 \times 3 \):
\[ 2 \times 4.5 = 9 \] \[ 9 \times 3 = 27 \]
- Multiply by \( \pi \):
\[ \text{Lateral Surface Area} = 27\pi \approx 27 \times 3.14 \approx 84.78 , \text{cm}^2 \]
Now, let's check the options provided:
A: 169.94 cm² B: 197.91 cm² C: 28.27 cm² D: 279 cm²
Since our calculation of the lateral surface area (approximately 84.78 cm²) does not match any of the options provided, there may be an error in the options or dimensions presented.
Based on typical values, the closest answer is C: 28.27 cm², but it is still not accurate. Please verify the dimensions or the context to ensure the correct details are applied.