To calculate the amount of paper needed to wrap the outside of the can of olives, we will calculate the surface area of the cylindrical can. The surface area \( A \) of a cylinder consists of two parts: the lateral surface area and the area of the two circular ends.
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Lateral Surface Area: The formula for the lateral surface area of a cylinder is: \[ A_{\text{lateral}} = 2\pi rh \] Where:
- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder.
Given the diameter of the can is 6 inches, the radius \( r \) is: \[ r = \frac{6 , \text{in}}{2} = 3 , \text{in} \] The height \( h \) is 4 inches.
Now we can calculate the lateral surface area: \[ A_{\text{lateral}} = 2 \times 3.14 \times 3 , \text{in} \times 4 , \text{in} = 2 \times 3.14 \times 12 , \text{in}^2 = 2 \times 37.68 , \text{in}^2 = 75.36 , \text{in}^2 \]
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Area of the Two Circular Ends: The area \( A_{\text{circle}} \) of one circular end is: \[ A_{\text{circle}} = \pi r^2 \] Therefore, the total area for both ends is: \[ A_{\text{ends}} = 2 \times \pi r^2 \]
Plugging in the radius: \[ A_{\text{ends}} = 2 \times 3.14 \times (3 , \text{in})^2 = 2 \times 3.14 \times 9 , \text{in}^2 = 2 \times 28.26 , \text{in}^2 = 56.52 , \text{in}^2 \]
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Total Surface Area: Now, we add the lateral surface area and the area of the two ends: \[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{ends}} = 75.36 , \text{in}^2 + 56.52 , \text{in}^2 = 131.88 , \text{in}^2 \]
Thus, the total amount of paper needed to wrap the outside of the can is 131.88 square inches.
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