To find the surface area of a cylinder, you need to know both the radius of its base and its height. For a cylinder with a radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex], the surface area [tex]\( A \)[/tex] can be calculated using the formula:
[tex]\[
A = 2\pi r (h + r)
\][/tex]
Given:
- Radius [tex]\( r = 38 \)[/tex] mm
- Height [tex]\( h = 51 \)[/tex] mm
Substituting the given values into the formula, we have:
[tex]\[
A = 2\pi \times 38 \times (51 + 38)
\][/tex]
First, compute the sum inside the parentheses:
[tex]\[
51 + 38 = 89
\][/tex]
Next, multiply the radius by this sum and then by [tex]\( 2\pi \)[/tex]:
[tex]\[
2 \pi \times 38 \times 89 = 2 \times 38 \times 89 \pi
\][/tex]
Calculate [tex]\( 2 \times 38 \)[/tex]:
[tex]\[
2 \times 38 = 76
\][/tex]
Then multiply this result by 89:
[tex]\[
76 \times 89 = 6764
\][/tex]
Finally, multiply by [tex]\( \pi \)[/tex]:
[tex]\[
6764 \pi \, \text{mm}^2
\][/tex]
Therefore, the surface area of the cylinder is:
[tex]\[
6764 \pi \, \text{mm}^2
\][/tex]
The correct answer is:
[tex]\[
6764 \pi \, \text{mm}^2
\][/tex]
