To calculate the surface area of a cylinder, we will use the formula:
\[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \]
Step 1: Calculate the areas of the top and bottom circles
The area of a circle is given by the formula:
\[ A = \pi r^2 \]
Since both the top and bottom are circles with the same radius (r = 9.7 yd):
\[ A_{\text{top}} = A_{\text{bottom}} = \pi (9.7)^2 \] Using \(\pi \approx 3.14\):
\[ A_{\text{top}} = A_{\text{bottom}} = 3.14 \times (9.7)^2 \] Calculating \( (9.7)^2 \):
\[ (9.7)^2 = 94.09 \]
Now calculate the area:
\[ A \approx 3.14 \times 94.09 \approx 295.73 \text{ yd}^2 \] Thus,
\[ A_{\text{top}} \approx 295.73 \text{ yd}^2 \] \[ A_{\text{bottom}} \approx 295.73 \text{ yd}^2 \]
Step 2: Calculate the area of the side (curved surface area)
First, we need to find the circumference of the top circle, which is given by:
\[ C = 2\pi r \] Substituting \( r = 9.7 \) yd:
\[ C = 2 \times 3.14 \times 9.7 \]
Calculating:
\[ C \approx 2 \times 3.14 \times 9.7 \approx 60.12 \text{ yd} \]
Now, the area of the side (curved surface area) is given by:
\[ A_{\text{side}} = C \times h \] Where \( h = 5.4 \) yd. Thus:
\[ A_{\text{side}} = 60.12 \times 5.4 \]
Calculating:
\[ A_{\text{side}} \approx 324.65 \text{ yd}^2 \]
Step 3: Calculate the total surface area
Adding all areas together:
\[ \text{Surface Area} = A_{\text{top}} + A_{\text{bottom}} + A_{\text{side}} \] \[ \text{Surface Area} \approx 295.73 + 295.73 + 324.65 \] \[ \text{Surface Area} \approx 916.11 \text{ yd}^2 \]
Final Answer:
The surface area of the cylinder is approximately 916.11 square yards.