To find the surface area of the cylinder, we will calculate the areas of the top, bottom, and curved surface using the given radius and height.
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Given Values:
- Radius (r) = 5.1 mm
- Height (h) = 4 mm
- Use \( \pi \approx 3.14 \)
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Calculate the Area of the Top and Bottom Circles: \[ A_{\text{top}} = A_{\text{bottom}} = \pi r^2 \] \[ A_{\text{top}} = A_{\text{bottom}} = 3.14 \times (5.1)^2 \] \[ = 3.14 \times 26.01 \approx 81.77 \text{ mm}^2 \]
Since there are two circles (top and bottom): \[ A_{\text{top}} + A_{\text{bottom}} = 81.77 + 81.77 = 163.54 \text{ mm}^2 \]
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Calculate the Circumference of the Top Circle: \[ C = 2 \pi r \] \[ = 2 \times 3.14 \times 5.1 \] \[ \approx 31.94 \text{ mm} \]
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Calculate the Area of the Curved Surface:
- The curved surface can be represented as a rectangle where one side is the height and the other side is the circumference of the circle. \[ A_{\text{curved}} = C \times h \] \[ = 31.94 \times 4 \] \[ \approx 127.76 \text{ mm}^2 \]
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Total Surface Area of the Cylinder: \[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{curved}} \] \[ = 163.54 + 127.76 \] \[ = 291.30 \text{ mm}^2 \]
Finally, rounding to the nearest hundredth, the surface area of the cylinder is approximately 291.30 square millimeters.