91Ó°ÊÓ

For the point \((r, \theta), r\) is the ________ ________ from \(O\) to \(P\) and \(\theta\) is the ________ ________ , counterclockwise from the polar axis to the line segment \(\overline{OP}\).

The graph of a hyperbola has two disconnected parts called ________.

A collection of points satisfying a geometric property can also be referred to as a ________ of points.

A curve traced by a point on the circumference of a circle as the circle rolls along a straight line in a plane is called a ________.

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(-2, \dfrac{2\pi}{3}\right)\)

SOUND LOCATION You and a friend live 4 miles apart (on the same "east-west" street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.)

In Exercises 39-54, find a polar equation of the conic with its focus at the pole. \(\textit{Conic}\) Hyperbola \(\textit{Vertex or Vertices}\) \((4, \pi/2), (1, \pi/2)\)

PROJECTILE MOTION A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_0\) feet per second, and the path of the projectile is modeled by the parametric equations \(x= (v_0 \cos\ \theta)t \quad \textrm{and} \quad y=h+(v_0 \sin\ \theta)t - 16t^2.\) In Exercises 61 and 62, use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\theta\) and \(v_0\). For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta = 60^{\circ}, \quad v_0 = 88\) feet per second (b) \(\theta = 60^{\circ}, \quad v_0 = 132\) feet per second (c) \(\theta = 45^{\circ}, \quad v_0 = 88\) feet per second (d) \(\theta = 45^{\circ}, \quad v_0 = 132\) feet per second

SATELLITE TRACKING A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately \(17,500\) miles per hour. If this velocity is multiplied by \(\sqrt{2}\), the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of Earth as the focus (see figure). (a) Find a polar equation of the parabolic path of the satellite (assume the radius of Earth is \(4000\) miles). (b) Use a graphing utility to graph the equation you found in part (a). (c) Find the distance between the surface of the Earth and the satellite when \(\theta\ =\ 30^{\circ}\). (d) Find the distance between the surface of Earth and the satellite when \(\theta\ =\ 60^{\circ}\).

PROJECTILE MOTION Eliminate the parameter \(t\) from the parametric equations \(x = (v_0 \cos\ \theta)t \quad \quad\) and \(\quad \quad y=h+(v_0 \sin\ \theta)t-16t^2\) and for the motion of a projectile to show that the rectangular equation is \(y=-\dfrac{16\ \sec^2\ \theta}{v_{0}^{2}} x^{2} + (\tan\ \theta)x + h\).