To calculate the total surface area of the cylindrical storage box, we will follow these steps using the dimensions provided: a radius of 4.5 inches and a height of 10.2 inches.
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Calculate the Area of One Circular Face: The formula for the area of a circle is: \[ A = \pi r^2 \] where \( r \) is the radius.
Given \( r = 4.5 \) inches: \[ A = 3.14 \times (4.5)^2 = 3.14 \times 20.25 = 63.615 \text{ square inches} \]
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Calculate the Total Area of the Circular Faces: Since there are two circular faces (top and bottom), we multiply the area of one face by 2: \[ \text{Total Area of Circular Faces} = 2 \times 63.615 = 127.23 \text{ square inches} \]
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Calculate the Circumference of the Circle: The formula for circumference \( C \) is: \[ C = 2\pi r \] Given \( r = 4.5 \) inches: \[ C = 2 \times 3.14 \times 4.5 = 28.26 \text{ inches} \]
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Calculate the Area of the Lateral Face: The formula for the lateral surface area of a cylinder is: \[ A_{\text{lateral}} = C \times h \] where \( h \) is the height of the cylinder. Given \( h = 10.2 \) inches: \[ A_{\text{lateral}} = 28.26 \times 10.2 = 288.652 \text{ square inches} \]
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Calculate the Total Surface Area of the Cylinder: The total surface area \( A_{\text{total}} \) is the sum of the area of the circular faces and the area of the lateral face: \[ A_{\text{total}} = \text{Total Area of Circular Faces} + A_{\text{lateral}} \] \[ A_{\text{total}} = 127.23 + 288.652 = 415.882 \text{ square inches} \]
Now we summarize the findings:
- Radius of circle: 4.5 inches
- Area of ONE Circular Face: 63.615 square inches
- Total Area of the Circular Faces: 127.23 square inches
- Circumference of Circle: 28.26 inches
- Height of Cylinder: 10.2 inches
- Area of the Lateral Face: 288.652 square inches
- Total Surface Area of Cylinder: 415.882 square inches
Thus, Simone will need approximately 415.882 square inches of patterned paper to cover the entire box.