Exam 10: Parametric Equations and Polar Coordinates

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at $A$ and $B$ transmit simultaneous signals to a ship or an aircraft located at $P$ . The onboard computer converts the time difference in receiving these signals into a distance difference $| A | - | B |$ , and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station $B$ is located $\mathrm { L } = 480$ mi due east of station $A$ on a coastline. $\mathrm { A }$ ship received the signal from $B 1280$ microseconds $( \mu s )$ before it received the signal from $A$ . Assuming that radio signals travel at a speed of $1000 \mathrm { ft } / \mu \mathrm { s }$ and if the ship is due north of $B$ , how far off the coastline is the ship? Round your answer to the nearest mile. Select the correct answer.

Describe the motion of a particle with position $( x , y )$ as $t$ varies in the given interval $0 \leq t \leq 2 \pi$ . $x = 8 \sin t , y = 5 \cos t$

If $a$ and $b$ are fixed numbers, find parametric equations for the set of all points $P$ determined as shown in the figure, using the angle ang as the parameter. Write the equations for $a = 12$ and $b = 4$ .

Find a polar equation for the curve represented by the given Cartesian equation. $x ^ { 2 } = 3 y$

Find the slope of the tangent line to the given polar curve at the point specified by the value of $a$ . $r = \frac { 1 } { a } , a = \pi$

$\text { Consider the polar equation } r = \frac { 7 } { 5 + 5 \cos \theta } \text {. }$ (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Find an equation of the tangent line to the curve at the point corresponding to the value of the parameter. $x = e ^ { \sqrt { t } } , \quad y = t - \ln t ^ { 6 } ; \quad t = 1$

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $0.8$ and vertex at $\left( 1 , \frac { \pi } { 2 } \right)$ .

$\text { Consider the polar equation } r = - \frac { 13 } { 4 \sin \theta - 1 } \text {. }$ (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Find the area of the region enclosed by one loop of the curve. Select the correct answer. $r = 7 \cos 8 \theta$

Eliminate the parameter to find a Cartesian equation of the curve. $x ( t ) = 2 \cos ^ { 2 } t , \quad y ( t ) = 7 \sin ^ { 2 } t$

The graph of the following curve is given. Find the area that it encloses. $r = 3 + 15 \sin 6 \theta$

Find the exact area of the surface obtained by rotating the given curve about the $x$ -axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = t \cos t , \quad y = t \sin t , \quad t = 5 \pi$

The planet Mercury travels in an elliptical orbit with eccentricity $0.703$ . Its minimum distance from the Sun is $8 \times 10 ^ { 7 } \mathrm {~km}$ . If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in km) from Mercury to the Sun.

The orbit of Hale-Bopp comet, discovered in 1995 , is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $366.5 \mathrm { AU }$ . [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ .) Find the answer in $\mathrm { AU }$ and round to the nearest hundredth.

A cow is tied to a silo with radius 9 by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow. Round the answer to the nearest hundredth.

Find an equation for the conic that satisfies the given conditions. parabola, vertex $( 0,0 )$ , focus $( 0 , - 4 )$

Find the eccentricity of the conic. $r = \frac { 5 } { 8 - 5 \sin \theta }$