Chapter 14.3, Problem 28E

Carnival rides

28. Suppose the carnival ride in Exercise 27 is modified so that Andrea’s position P (in ft) at time t (in s) is

$r(t)=\langle 20\cos t+10\cos 5t,20\sin t+10\sin 5t,5\sin 2t\rangle$.

a. Describe how this carnival ride differs from the ride in Exercise 27.

b. Find the speed function |v(t)| = υ(t) and plot its graph.

c. Find Andrea’s maximum and minimum speeds.

27 Consider a carnival ride where Andrea is at point P that moves counterclockwise around a circle centered at C while the arm, represented by the line segment from the origin O to point C, moves counterclockwise about the origin (see figure). Andrea's position (in feet) at time t (in seconds) is

$r(t)=\langle 20\cos t+10\cos 5t,20\sin t+10\sin 5t\rangle$

a. Plot a graph of r(t), for 0 ≤ t ≤ 2π.

b. Find the velocity v(t).

c. Show that the speed $\text{|}v(t)|=\upsilon (t)=10\sqrt{29+20\cos 4t}$and plot the speed, for 0 ≤ t ≤ 2π.(Hint: Use the identity sin mx sin nx + cos mx cos nx = cos((m – n)x).)

d. Determine Andrea’s maximum and minimum speeds.