To find the total surface area of the cylindrical storage box, we will first need the required dimensions and then apply the formulas for the surface area of a cylinder.
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Radius of Circle: The problem states that circles are 4.5 inches. Therefore, the diameter of the circle is 4.5 inches. To find the radius (r): \[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{4.5}{2} = 2.25 , \text{inches} \]
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Area of Each Circular Face: The area of a circle is given by the formula: \[ \text{Area} = \pi r^2 \] Using \(\pi \approx 3.14\): \[ \text{Area of Each Circular Face} = 3.14 \times (2.25)^2 = 3.14 \times 5.0625 = 15.90 , \text{square inches} \]
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Total Area of the Circular Faces: Since there are two circular faces (top and bottom): \[ \text{Total Area of the Circular Faces} = 2 \times 15.90 = 31.80 , \text{square inches} \]
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Circumference of Circle: The circumference (C) of a circle is calculated by: \[ C = 2\pi r \] Substituting the values: \[ C = 2 \times 3.14 \times 2.25 = 14.13 , \text{inches} \]
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Height of Cylinder: The problem states the sides of squares are 10.2 inches, and assuming that represents the height of the cylinder: \[ \text{Height of Cylinder} = 10.2 , \text{inches} \]
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Area of the Lateral Face: The area of the lateral surface of a cylinder is given by: \[ \text{Area of Lateral Face} = C \times \text{Height} \] Plugging in the values: \[ \text{Area of Lateral Face} = 14.13 \times 10.2 = 144.90 , \text{square inches} \]
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Total Surface Area of the Cylinder: The total surface area (SA) of the cylinder is the sum of the areas of the circular faces and the lateral face: \[ \text{Total Surface Area} = \text{Total Area of Circular Faces} + \text{Area of Lateral Face} \] \[ \text{Total Surface Area} = 31.80 + 144.90 = 176.70 , \text{square inches} \]
Summary of the Results:
- Radius of Circle = 2.25 inches
- Area of Each Circular Face = 15.90 square inches
- Total Area of the Circular Faces = 31.80 square inches
- Circumference of Circle = 14.13 inches
- Height of Cylinder = 10.2 inches
- Area of the Lateral Face = 144.90 square inches
- Total Surface Area of Cylinder = 176.70 square inches
Simone will need 176.70 square inches of patterned paper to cover her cylindrical storage box.