A $20-\mathrm{ft}$ Ferris wheel turns counterclockwise one revolution every 12 sec (see figure). Eric stands at point $D, 75 \mathrm{ft}$ from the base of the wheel. At the instant Jane is at point $A,$ Eric throws a ball at the Ferris wheel, releasing it from the same height as the bottom of the wheel. If the ball's initial speed is 60 $\mathrm{ft} / \mathrm{sec}$ and it is released at an angle of $120^{\circ}$ with the horizontal, does Jane have a chance to catch the ball? Follow the steps below to obtain the answer.
(a) Assign a coordinate system so that the bottom car of the Ferris wheel is at $(0,0)$ and the center of the wheel is at $(0,20) .$ Then Eric releases the ball at the point $(75,0) .$ Explain why parametric equations for Jane's path are
$$x_{1}=20 \cos \left(\frac{\pi}{6} t\right), y_{1}=20+20 \sin \left(\frac{\pi}{6} t\right), t \geq 0$$
(b) Explain why parametric equations for the path of the ball are
$$x_{2}=-30 t+75, y_{2}=-16 t^{2}+(30 \sqrt{3}) t, t \geq 0$$
(c) Graph the two paths simultaneously and determine if Jane and the ball arrive at the point of intersection of the two paths at the same time.
(d) Find a formula for the distance $d(t)$ between Jane and the ball at any time $t$
(e) Use the graph of the parametric equations $x_{3}=t, y_{3}=d(t),$ to estimate the minimum distance between Jane and the ball and when it occurs. Doyou think Jane has a chance to
catch the ball?