Answer: The angle between line AB and the axis of the cylinder is 60 degrees.
Explanation:
Let's draw a cross-sectional diagram of the cylinder to help visualize the problem.
Label the center of the top circle as O and the center of the bottom circle as O'.
Label the midpoint of line AB as M.
Draw a line from M to the center of the cylinder, which intersects the axis of the cylinder at point C.
Because line AB is perpendicular to the axis of the cylinder, line MC is also perpendicular to line AB.
Label the length of line MC as h, and the distance between point M and the axis of the cylinder as x.
By the Pythagorean theorem, we know that OM^2 + h^2 = R^2 (the radius of the cylinder)
Similarly, O'M^2 + h^2 = R^2
Subtracting these two equations, we get OM^2 - O'M^2 = 0, which means that OM = O'M = R.
Therefore, triangle MOC is an isosceles triangle with MO and O'M both equal to R.
Because x is the distance between line AB and the axis of the cylinder, we know that x = MC - (R√3)÷2.
We also know that h = R - x (because OM = R).
Using the Pythagorean theorem, we can solve for MC: (R^2 - h^2)^0.5 = MC
Substituting h = R - x, we get MC = (2Rx - x^2)^0.5
Setting MC = (R√3)÷2 (from the problem statement), we can solve for x: x = R(3 - 3^0.5)^0.5
Finally, using the tangent function, we can solve for the angle between line AB and the axis of the cylinder: tanθ = (R√3)÷2 / x, where θ is the angle we are looking for.
Substituting x from step 15, we get tanθ = 1 / (3 - 3^0.5)
Using a calculator, we can solve for θ: θ = 60 degrees.