To calculate the surface area of the label needed for the can of olives, we need to find the lateral surface area and the area of the top and bottom of the can.
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Lateral Surface Area (side of the can): The formula for the lateral surface area of a cylinder is: \[ \text{Lateral Surface Area} = 2 \pi r h \] where \( r \) is the radius and \( h \) is the height.
The diameter of the can is 6 inches, so the radius \( r \) is: \[ r = \frac{6}{2} = 3 \text{ inches} \] The height \( h \) is given as 4 inches.
Plugging in these values: \[ \text{Lateral Surface Area} = 2 \times 3.14 \times 3 \times 4 \] \[ = 2 \times 3.14 \times 12 \] \[ = 2 \times 37.68 \] \[ = 75.36 \text{ square inches} \]
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Area of the Top and Bottom: The area of one circle (top or bottom) is given by the formula: \[ \text{Area} = \pi r^2 \] For both the top and bottom, we double this: \[ \text{Area of Top and Bottom} = 2 \pi r^2 \] Plugging in the radius: \[ = 2 \times 3.14 \times (3^2) \] \[ = 2 \times 3.14 \times 9 \] \[ = 2 \times 28.26 \] \[ = 56.52 \text{ square inches} \]
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Total Surface Area (Label area needed): Now, we sum the lateral surface area and the area of the top and bottom: \[ \text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of Top and Bottom} \] \[ = 75.36 + 56.52 \] \[ = 131.88 \text{ square inches} \]
Therefore, the total area of paper needed for the label on the outside of the can is:
131.88 square inches.