The formula for the surface area of a cylinder is:
S = 2πrh + 2πr²
where S is the surface area, r is the radius, and h is the height.
We are given that the height of the cylinder is 12 yards and the circumference is 514.35 yards.
The circumference of a cylinder is given by the formula:
C = 2πr
where C is the circumference and r is the radius.
Substituting the given value of the circumference, we get:
514.35 = 2πr
r = 514.35 / (2π)
r ≈ 81.97
Now we can use the formula for surface area:
S = 2πrh + 2πr²
Substituting the given values, we get:
S = 2(3.14)(81.97)(12) + 2(3.14)(81.97)²
S ≈ 50,684.97
Therefore, the surface area of the cylinder is approximately 50,684.97 square yards.