To find the area of the label needed for the can of olives, we need to calculate the lateral surface area of the cylinder, as well as the area of the top and bottom circles.
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Lateral Surface Area (A_lateral): The formula for the lateral surface area of a cylinder is given by: \[ A_{lateral} = 2 \pi r h \] 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 4 inches.
Plugging in the values: \[ A_{lateral} = 2 \times 3.14 \times 3 \times 4 \] \[ A_{lateral} = 2 \times 3.14 \times 12 \] \[ A_{lateral} = 2 \times 37.68 = 75.36 \text{ square inches} \]
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Area of the Top and Bottom Circles (A_circles): The area of one circle is given by the formula: \[ A_{circle} = \pi r^2 \] Therefore, the area of both circles is: \[ A_{circles} = 2 \times \pi r^2 \] Plugging in the radius: \[ A_{circles} = 2 \times 3.14 \times 3^2 \] \[ A_{circles} = 2 \times 3.14 \times 9 \] \[ A_{circles} = 2 \times 28.26 = 56.52 \text{ square inches} \]
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Total Area (A_total): The total area of the label on the outside of the can is the sum of the lateral area and the area of the top and bottom circles: \[ A_{total} = A_{lateral} + A_{circles} \] \[ A_{total} = 75.36 + 56.52 = 131.88 \text{ square inches} \]
Therefore, the total area of paper needed for the label on the outside of the can is 131.88 square inches.
The correct answer is 131.88.