To find the surface area of the cylinder, we first need to calculate each component of the surface area formula:
\[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \]
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Calculate the area of the top circle \(A_{\text{top}}\): \[ A_{\text{top}} = \pi r^2 \] where \(r = 5.9 \text{ mm}\).
\[ A_{\text{top}} = 3.14 \times (5.9)^2 \approx 3.14 \times 34.81 \approx 109.94 \text{ mm}^2 \]
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Calculate the area of the bottom circle \(A_{\text{bottom}}\): Since the bottom circle is the same as the top circle: \[ A_{\text{bottom}} = A_{\text{top}} \approx 109.94 \text{ mm}^2 \]
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Calculate the circumference \(C\) of the top circle: \[ C = 2\pi r \] \[ C = 2 \times 3.14 \times 5.9 \approx 6.28 \times 5.9 \approx 37.05 \text{ mm} \]
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Calculate the area of the side \(A_{\text{side}}\) (which is the curved surface): The area of the side is the rectangle formed by the height and the circumference: \[ A_{\text{side}} = C \times h \] where \(h = 15 \text{ mm}\). \[ A_{\text{side}} = 37.05 \times 15 \approx 555.75 \text{ mm}^2 \]
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Now add up all the areas: \[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \] \[ \text{Surface Area} = 109.94 + 109.94 + 555.75 \approx 775.63 \text{ mm}^2 \]
Rounding this to the nearest hundredth, the surface area of the cylinder is approximately: \[ \text{Surface Area} \approx 775.63 \text{ mm}^2 \]
So, to summarize:
The surface area of the cylinder is about 775.63 square millimeters.