To calculate the total surface area of the cylindrical storage box, we need to determine the areas of its circular faces and the lateral surface area.
Step 1: Finding the radius and area of the circular faces
Given that circles have a diameter of 4.5 inches, we can find the radius:
- Radius of circle = \( \frac{\text{diameter}}{2} = \frac{4.5}{2} = 2.25 \) inches
Now we calculate the area of one circular face:
- Area of each Circular Face = \( \pi r^2 = 3.14 \times (2.25)^2 \)
Calculating \( (2.25)^2 \):
\[ (2.25)^2 = 5.0625 \]
Now calculate the area:
\[ \text{Area of each Circular Face} = 3.14 \times 5.0625 = 15.87375 \text{ square inches} \]
Now find the total area of the circular faces (there are two):
- Total Area of the Circular Faces = \( 2 \times \text{Area of each Circular Face} = 2 \times 15.87375 = 31.7475 \text{ square inches} \)
Step 2: Finding the circumference and lateral area
Next, we calculate the circumference of the base circle:
- Circumference of Circle = \( 2 \pi r = 2 \times 3.14 \times 2.25 \)
Calculating:
\[ 2 \times 3.14 = 6.28 \]
Then calculate:
\[ \text{Circumference of Circle} = 6.28 \times 2.25 = 14.115 \text{ inches} \]
Now that we have the circumference, we take the height of the cylinder, which is given as 10.2 inches.
Now, we can find the area of the lateral face of the cylinder:
- Area of the Lateral Face = \( \text{Circumference} \times \text{Height} = 14.115 \times 10.2 \)
Calculating the lateral area:
\[ 14.115 \times 10.2 = 144.453 \text{ square inches} \]
Step 3: Total Surface Area
To find the total surface area of the cylinder, we sum the area of the circular faces and the lateral area:
- Total Surface Area of Cylinder = \( \text{Total Area of the Circular Faces} + \text{Area of the Lateral Face} \)
Calculating:
\[ \text{Total Surface Area of Cylinder} = 31.7475 + 144.453 = 176.2005 \text{ square inches} \]
Summary of Results
- Radius of circle = \( 2.25 \) inches
- Area of each Circular Face = \( 15.87375 \) square inches
- Total Area of the Circular Faces = \( 31.7475 \) square inches
- Circumference of Circle = \( 14.115 \) inches
- Height of Cylinder = \( 10.2 \) inches
- Area of the Lateral Face = \( 144.453 \) square inches
- Total Surface Area of Cylinder = \( 176.2005 \) square inches
Thus, Simone will need approximately 176.2005 square inches of patterned paper to cover the entire box.