91Ó°ÊÓ

For Exercises \(41-48\), for each complex number \(z\), write the complex conjugate \(\bar{z}\), and find \(z \bar{z}\). $$ z=2-3 i $$

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r=-3 $$

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. Suppose the ride conductor was rather sinister and speeded up the ride to twice the speed. How would you modify the parametric equations to model such a change? Now vary the values of \(A\) and \(B\). What do you conjecture these parameters are modeling in this problem?

Fan Blade. The position on the tip of a ceiling fan is given by the parametric equations \(x=\sin (10 t)\) and \(y=\cos (10 t)\), where \(x\) and \(y\) are the vertical and lateral positions relative to the center of the fan, respectively, and \(t\) is the time in seconds. How long does it take for the fan, blade to make one complete revolution?

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2}=9 \cos (2 \theta) $$

In Exercises 45-60, express each complex number in exact rectangular form. $$ \frac{3}{2}\left[\cos \left(\frac{7 \pi}{6}\right)+i \sin \left(\frac{7 \pi}{6}\right)\right] $$

For Exercises 49-64, write each quotient in standard form. $$ \frac{2+3 i}{3-5 i} $$

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r=4 \theta $$

For Exercises 71 and 72, refer to the following: Spirals are seen in nature-for example, in the swirl of a pine cone. They are also used in machinery to convert motions. An Archimedes spiral has the general equation \(r=a \theta\). A more general form for the equation of a spiral is \(r=a \theta^{1 / n}\), where \(n\) is a constant that determines how tightly the spiral is wrapped. Archimedes Spiral. Compare the Archimedes spiral \(r=\theta\) with the spiral \(r=\theta^{1 / 2}\) by graphing both on the same polar graph.

For Exercises 73 and 74, refer to the following: The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with different wing profiles. Flapping Wings of Birds. Compare the two following possible lemniscate patterns by graphing them on the same polar graph: \(r^{2}=4 \cos (2 \theta)\) and \(r^{2}=4 \cos (2 \theta+2)\).