To find the surface area of a cylinder, we use the formula:
\[ \text{Surface Area} = 2\pi r(h + r) \]
where \( r \) is the radius, \( h \) is the height, and \( \pi \) is approximately 3.14.
First, we need to determine the radius from the diameter:
\[ \text{Diameter} = 8.3 , \text{mm} \quad \Rightarrow \quad r = \frac{\text{Diameter}}{2} = \frac{8.3}{2} = 4.15 , \text{mm} \]
Next, we can substitute the values of \( r \) and \( h \) into the surface area formula. The height \( h \) is given as 6.7 mm.
Using the surface area formula:
\[ \text{Surface Area} = 2\pi r(h + r) \]
Substituting the values:
\[ \text{Surface Area} = 2 \times 3.14 \times 4.15 \times (6.7 + 4.15) \]
Calculating \( (h + r) \):
\[ h + r = 6.7 + 4.15 = 10.85 \]
Now substituting this value back:
\[ \text{Surface Area} = 2 \times 3.14 \times 4.15 \times 10.85 \]
Calculating \( 2 \times 3.14 = 6.28 \):
\[ \text{Surface Area} = 6.28 \times 4.15 \times 10.85 \]
Calculating \( 6.28 \times 4.15 \):
\[ 6.28 \times 4.15 \approx 26.052 \]
Now, multiply this result by 10.85:
\[ 26.052 \times 10.85 \approx 282.2792 \]
Finally, rounding to the nearest hundredth:
\[ \text{Surface Area} \approx 282.28 , \text{mm}^2 \]
Thus, the surface area of the cylinder is approximately 282.28 mm².