Answer: D

Small Hint:

Place the center at height $20$; the rider rises $10$ feet from the bottom

Big Hint:

The rider is now $10$ feet below center; find the central angle turned from the bottom

Solution:

Put the center $O$ at height $20.$ The bottom $A$ is at height $0,$ and the rider reaches height $10,$ which is $10$ feet below the center.

Draw a horizontal segment from the rider to the wheel’s vertical diameter. The resulting right triangle has a vertical leg of length $10$ and a hypotenuse (the radius) of length $20.$ That leg is half the hypotenuse, so the radius to the rider makes $60^\circ$ with the downward vertical.

The wheel turns $360^\circ$ in $60$ seconds, so turning $60^\circ$ takes $\dfrac{60}{360}\cdot 60 = 10$ seconds.

Thus, the correct answer is D.