Exam 10: Parametric Equations and Polar Coordinates

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.

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

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 $\mathrm { km }$ ) from Mercury to the Sun. Select the correct answer.

The graph of the following curve is given. Find the area that it encloses.

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

Find the equation of the directrix of the conic. $r = \frac { 6 } { 3 + \sin \theta }$

Match the equation with the correct graph. $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$

Find the vertex, focus, and directrix of the parabola. $y ^ { 2 } - 2 y - 20 x + 81 = 0$

Find an equation for the conic that satisfies the given conditions. ellipse, foci $( \pm 1,6 )$ , length of major axis 8

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

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

If a projectile is fired with an initial velocity of $v _ { 0 }$ meters per second at an angle $\alpha$ above the horizontal and air resistance is assumed to be negligible, then its position after $t$ seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t , y = \left( v _ { 0 } \sin \alpha \right) t - \frac { 1 } { 2 } g t ^ { 2 }$ where $g$ is the acceleration of gravity $\left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right)$ . If a gun is fired with $\alpha = 55 ^ { \circ }$ and $v _ { 0 } = 440 \mathrm {~m} / \mathrm { s }$ when will the bullet hit the ground?

Find an equation for the conic that satisfies the given conditions. hyperbola, foci $( 0 , \pm 6 )$ , vertices $( 0 , \pm 3 )$

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 7 } { 2 }$ , and directrix $y = - 7$ . Identify the conic.

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 $( \mu s )$ 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.

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 7 } { 2 }$ , and directrix $y = - 7$ . Identify the conic.

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.