Chapter 12, Problem 14PS

Ferris Wheel You want to toss an object to a friend who is riding a Ferris wheel (see figure). The following parametric equations give the path of the friend $r_1(t)$ and the path of the object $r_2(t)$. Distance is measured in meters, and time is measured in seconds.

$\begin{array}{c}{r}_1(t)=15\left(\sin \frac{\pi t}{10}\right)i+\left(16-15\cos \frac{\pi t}{10}\right)j\\ \begin{array}{l}{r}_2(t)=[22-8.03(t-t_0)]i+\\ [1+11.47(t-t_0)-4.9{(}^t]j\end{array}\end{array}$

(a) Locate your friend’s position on the Ferris wheel at time $t=0$.

(b) Determine the number of revolutions per minute of the Ferris wheel.

(c) What are the speed and angle of inclination (in degrees) at I which the object is thrown at time $t=t_0?$

(d) Use a graphing utility to graph the vector-valued functions using a value of $t_0$ that allows your friend to be within reach of the object. (Do this by trial and error.) Explain the significance of $t_0$.

(e) Find the approximate time your friend should be able to catch the object. Approximate the speeds of your friend and the object at that time.