To calculate the area needed for the label on the outside of the can, we need to find the surface area of the label, which includes the lateral surface area and the area of the two circular ends (top and bottom).
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Calculate the radius of the can: \[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6 \text{ inches}}{2} = 3 \text{ inches} \]
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Calculate the lateral surface area (the side of the can): \[ \text{Lateral Surface Area} = 2 \pi r h \] Where \( r = 3 \) inches and \( h = 4 \) inches. Plugging in the values: \[ \text{Lateral Surface Area} = 2 \times 3.14 \times 3 \times 4 = 2 \times 3.14 \times 12 = 75.36 \text{ square inches} \]
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Calculate the area of the top and bottom (2 circles): \[ \text{Area of one circle} = \pi r^2 = 3.14 \times (3^2) = 3.14 \times 9 = 28.26 \text{ square inches} \] Since there are two circles: \[ \text{Area of both circles} = 2 \times 28.26 = 56.52 \text{ square inches} \]
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Total surface area for the label: \[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of both circles} \] \[ \text{Total Surface Area} = 75.36 + 56.52 = 131.88 \text{ square inches} \]
So, the total area of paper needed for the label on the outside of the can is 131.88 square inches.
The answer is 131.88.