## 67. Throwing a Ball at a Ferris Wheel A 20−ft
Ferris wheel turns counterclockwise one revolution every 12 sec (see figure). Eric stands at point D,75ft 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 60ft/sec and it is released at an angle of 120∘ 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
x1=20cos(6πt),y1=20+20sin(6πt),t≥0.
(b) Explain why parametric equations for the path of the ball are
x2=−30t+75,y2=−16t2+(303)t,t≥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) Writing to Learn Use the graph of the parametric equations x3=t,y3=d(t), to estimate the minimum distance between Jane and the ball and when it occurs. Do you think Jane has a chance to catch the ball?
