Exam 10: Parametric Equations and Polar Coordinates

$\text { Find } \frac { d ^ { 2 } y } { d x ^ { 2 } } \text {. }$ $x = 5 + t ^ { 2 } , y = t - t ^ { 3 }$

Find the vertices, foci and asymptotes of the hyperbola. $5 x ^ { 2 } - 5 y ^ { 2 } + 40 x - 50 y = 70$

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pi$

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$ .

Find the area of the region that is bounded by the given curve and lies in the specified sector. $r = 11 \sqrt { \sin 2 \theta } , 0 \leq \theta \leq \frac { \pi } { 2 }$

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 $\mathrm { L } = 480 \mathrm { mi }$ due east of station $A$ on a coastline. A ship received the signal from $B 1280$ microseconds $( \mu \mathrm { s } )$ before it received the signal from $A$ . Assuming that radio signals travel at a speed of $1000 \mathrm { ft } / \mu s$ 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.

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , \quad r = 1 + \cos \theta$

If the parametric curve $x = f ( t ) , \quad y = g ( t )$ satisfies $g ^ { \prime } ( 4 ) = 0$ , then it has a horizontal tangent when $t = 4$ .

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. $x = 4 t \cos t , y = 4 t \sin t , t = - \pi$

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

Find the area of the region that lies inside the first curve and outside the second curve. $r = 3 \cos \theta , r = 1 + \cos \theta$

Set up, but do not evaluate, an integral that represents the length of the parametric curve. $x = t - t ^ { 10 } , y = \frac { 10 } { 9 } t ^ { 9 / 8 } , \quad 8 \leq t \leq 18$

Find an equation of the ellipse that satisfies the given conditions. Foci: $( 0 , \pm 8 )$ , vertices $( 0 , \pm 9 )$

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$

Eliminate the parameter to find a Cartesian equation of the curve. $x = e ^ { 4 t } - 5 , \quad y = e ^ { 8 t }$

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 AU and round to the nearest hundredth.

Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

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