To find the surface area of the cylinder, we can use the formula:
\[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \]
Where:
- \( A_{\text{top}} \) and \( A_{\text{bottom}} \) are the areas of the circular top and bottom, respectively.
- \( A_{\text{side}} \) is the area of the curved surface of the cylinder.
Step 1: Calculate the Area of the Top and Bottom Circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Where:
- \( r \) is the radius.
Given \( r = 3.2 \) inches:
\[ A_{\text{top}} = A_{\text{bottom}} = \pi (3.2)^2 \]
Calculating:
\[ A_{\text{top}} = A_{\text{bottom}} = 3.14 \times (3.2)^2 = 3.14 \times 10.24 \approx 32.24 , \text{square inches} \]
Since we have two identical circles:
\[ A_{\text{top}} + A_{\text{bottom}} = 32.24 + 32.24 = 64.48 , \text{square inches} \]
Step 2: Calculate the Circumference of the Top Circle
The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Calculating:
\[ C = 2 \times 3.14 \times 3.2 \approx 19.88 , \text{inches} \]
Step 3: Calculate the Area of the Curved Surface
The area of the curved surface \( A_{\text{side}} \) can be viewed as a rectangle with one side being the height \( h \) and the other side being the circumference \( C \):
\[ A_{\text{side}} = C \times h \]
Given \( h = 2.3 \) inches:
\[ A_{\text{side}} = 19.88 \times 2.3 \approx 45.70 , \text{square inches} \]
Step 4: Calculate the Total Surface Area
Now, add the areas together to find the total surface area of the cylinder:
\[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} = 64.48 + 45.70 \approx 110.18 , \text{square inches} \]
Final Answer
The surface area of the cylinder is approximately \( \boxed{110.18} \) square inches.
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