Deck 10: Parametric Equations and Polar Coordinates

Question 1

Find parametric equations for the path of a particle that moves once clockwise along the circle starting at

Answer 1

Question 2

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune.The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude and apolune altitude (above the moon).Find an equation of this ellipse if the radius of the moon is and the center of the moon is at one focus.

Answer 2

Question 3

Set up, but do not evaluate, an integral that represents the length of the parametric curve.

Answer 3

Question 4

Find the area enclosed by the curve

Question 5

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

Question 6

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 7

True or False? The exact length of the parametric curve

Question 8

If and are fixed numbers, find parametric equations for the set of all points etermined as shown in the figure, using the angle as the parameter.Write the equations for and

Question 9

True or False? If the parametric curve satisfies then it has a horizontal tangent when

Question 10

Find an equation of the tangent to the curve at the point by first eliminating the parameter.

Question 11

Find the area that the curve encloses.

Question 12

Find the area of the region that lies inside both curves.

Question 13

Find the area bounded by the curve and the line

Question 14

Question 15

Set up an integral that represents the length of the curve.Then use your calculator to find the length correct to four decimal places.

Question 16

Eliminate the parameter to find a Cartesian equation of the curve.

Question 17

Eliminate the parameter to find a Cartesian equation of the curve.

Question 18

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 19

Using the arc length formula, set up, but do not evaluate, an integral equal to the total arc length of the ellipse.

Question 20

Question 21

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

Question 22

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune.The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude and apolune altitude (above the moon).Find an equation of this ellipse if the radius of the moon is and the center of the moon is at one focus.

Question 23

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 24

Find the area of the region that is bounded by the given curve and lies in the specified sector.

Question 25

Find the vertices, foci, and asymptotes of the hyperbola.

Question 26

Find the vertex, focus, and directrix of the parabola.

Question 27

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Question 28

Consider the polar equation (a) Find the eccentricity and an equation of the directrix of the conic.(b) Identify the conic.(c) Sketch the curve.

Question 29

Find the vertex, focus, and directrix fo the parabola.

Question 30

Eliminate the parameter to find a Cartesian equation of the curve.

Question 31

Find an equation of the conic satisfying the given conditions.Hyperbola, foci (5, 6) ptotes x = 2y + 3 and

Question 32

Find a Cartesian equation for the curve described by the given polar equation.

Question 33

True or False? The exact length of the parametric curve

Question 34

True or False? If the parametric curve satisfies then it has a horizontal tangent when

Question 35

Find an equation of the conic satisfying the given conditions.Hyperbola, foci (5, 6) ptotes x = 2y + 1 and

Question 36

Question 37

Consider the polar equation (a) Find the eccentricity and an equation of the directrix of the conic.(b) Identify the conic.(c) Sketch the curve.

Question 38

Eliminate the parameter to find a Cartesian equation of the curve.

Question 39

Consider the polar equation (a) Find the eccentricity and an equation of the directrix of the conic.(b) Identify the conic.(c) Sketch the curve.

Question 40

Find an equation of the tangent line to the curve at the point corresponding to the value of the parameter.

Question 41

Select the correct Answer: for each question.
Find the length of the polar curve.

A)
B)
C)
D)
E)None of these

Question 42

Select the correct Answer: for each question.
Find an equation of the hyperbola centered at the origin that satisfies the given condition. ptotes:

A)
B)
C)
D)

Question 43

Select the correct Answer: for each question.
Find the exact area of the surface obtained by rotating the given curve about the x-axis.

A)
B)
C)
D)
E)None of these

Question 44

Select the correct Answer: for each question.
Find a polar equation for the curve represented by the given Cartesian equation.

A)
B)
C)
D)
E)

Question 45

Select the correct Answer: for each question.
A cross-section of a parabolic reflector is shown in the figure.The bulb is located at the focus and the opening at the focus is 18 cm.Find an equation of the parabola.Let be the origin.Find the diameter of the opening cm from the vertex.

A)
B)
C)The equation is
D)
E)The equation is
F)The equation is

Question 46

Select the correct Answer: for each question.
Write a polar equation of the conic that has a focus at the origin, eccentricity and directrix Identify the conic.

A) hyperbola
B) hyperbola
C) ellipse
D) ellipse

Question 47

Select the correct Answer: for each question.
Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and directrix

A)
B)
C)
D)
E)

Question 48

Select the correct Answer: for each question.
Write a polar equation in and of an ellipse with the focus at the origin, with the eccentricity and directrix

A)
B)
C)
D)
E)

Question 49

Select the correct Answer: for each question.
Find the point(s) of intersection of the curves and

A)
B)
C)
D)
E)

Question 50

Select the correct Answer: for each question.
Find an equation for the conic that satisfies the given conditions.hyperbola, , vertices (0, ±

A)
B)
C)
D)
E)

Question 51

Select the correct Answer: for each question.
Use a graph to estimate the values of for which the curves and intersect.Round yourAnswer to two decimal places.

A)
B)
C)
D)
E)

Question 52

Select the correct Answer: for each question.
Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus.Its perihelion distance (minimum distance from the planet to the sun) is approximately km, and its aphelion distance (maximum distance from the planet to the sun) is approximately km.Approximate the eccentricity of the planet's orbit.Round to three decimal places.

A)1.174
B)12.5
C)0.08
D)0.852

Question 53

Select the correct Answer: for each question.
The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity and one focus at the Sun.The length of its major axis is 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 and the aphelion distance is ) Find theAnswer in AU and round to the nearest hundredth.

A)
B)
C)
D)
E)

Question 54

Select the correct Answer: for each question.
Find the point(s) on the curve where the tangent is horizontal.

A)
B)
C)
D)
E)None of these

Question 55

Select the correct Answer: for each question.
Find parametric equations to represent the line segment from

A)
B)
C)
D)
E)

Question 56

Select the correct Answer: for each question.
Find an equation of the ellipse that satisfies the given conditions.Foci: (0, ± 8), vertices

A)
B)
C)
D)

Question 57

Select the correct Answer: for each question.
Find the vertices, foci and asymptotes of the hyperbola.

A)The foci are
B)The vertices are
C)The foci are
D)The vertices are
E)The asymptote is
F)The asymptotes are

Question 58

Select the correct Answer: for each question.
Find the eccentricity of the conic.

A)
B)
C)
D)
E)

Question 59

Select the correct Answer: for each question.
The planet Mercury travels in an elliptical orbit with eccentricity Its minimum distance from the Sun is km.If the perihelion distance from a planet to the Sun is and the aphelion distance is , find the maximum distance (in km) from Mercury to the Sun.

A) km
B) km
C) km
D) km
E) km

Question 60

Select the correct Answer: for each question.
The graph of the following curve is given.Find the area that it encloses.

A)
B)
C)
D)
E)

Question 61

Select the correct Answer: for each question
Find an equation of the parabola with focus and directrix

A)
B)
C)
D)
E)

Question 62

Select the correct Answer: for each question
Find an equation for the conic that satisfies the given conditions.ellipse, foci length of major axis 8

A)
B)
C)
D)
E)

Question 63

Select the correct Answer: for each question
Write a polar equation in r and of an ellipse with the focus at the origin, with the eccentricity and vertex at

A)
B)
C)
D)
E)

Question 64

Select the correct Answer: for each question
If a projectile is fired with an initial velocity of meters per second at an angle above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations where g is the acceleration of gravity If a gun is fired with and when will the bullet hit the ground?

A)
B)
C)
D)
E)

Question 65

Select the correct Answer: for each question
Sketch the polar curve with the given equation.

A)
B)
C)
D)

Question 66

Select the correct Answer: for each question
Find the area of the region that lies inside the first curve and outside the second curve.

A)
B)
C)
D)
E)

Question 67

Select the correct Answer: for each question
Find the length of the polar curve.

A)
B)
C)
D)
E)None of these

Question 68

Select the correct Answer: for each question
Describe the motion of a particle with position as varies in the given interval

A)Moves once counterclockwise along the circle starting and ending at
B)Moves once counterclockwise along the ellipse starting and ending at
C)Moves once counterclockwise along the ellipse starting and ending at
D) Moves once clockwise along the ellipse starting and ending at
E)Moves once clockwise along the circle starting and ending at

Question 69

Select the correct Answer: for each question
Find a polar equation for the curve represented by the given Cartesian equation.

A)
B)
C)
D)
E)

Question 70

Select the correct Answer: for each question
Find the polar equation for the curve represented by the given Cartesian equation.

A)
B)
C)
D)
E)

Question 71

Select the correct Answer: for each question
Write a polar equation in nd f an ellipse with the focus at the origin, with the eccentricity and directrix

A)
B)
C)
D)
E)

Question 72

Select the correct Answer: for each question
The curve cross itself at some point Find the equations of both tangent lines at that point.

A)
B)
C)
D)
E)

Question 73

Select the correct Answer: for each question
Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus.Its perihelion distance (minimum distance from the planet to the sun) is approximately km, and its aphelion distance (maximum distance from the planet to the sun) is approximately km.Approximate the eccentricity of the planet's orbit.Round to three decimal places.

A)1.174
B)12.5
C)0.08
D)0.852

Question 74

Select the correct Answer: for each question
In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at and transmit simultaneous signals to a ship or an aircraft located at The onboard computer converts the time difference in receiving these signals into a distance difference 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 is located mi due east of station on a coastline.A ship received the signal from microseconds efore it received the signal from .Assuming that radio signals travel at a speed of and if the ship is due north of , how far off the coastline is the ship? Round yourAnswer to the nearest mile.

A)
B)
C)
D)
E)

Question 75

Select the correct Answer: for each question
Find the exact area of the surface obtained by rotating the given curve about the x-axis.

A)
B)
C)
D)
E)None of these

Question 76

Select the correct Answer: for each question
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

A)
B)
C)
D)
E)None of these

Question 77

Select the correct Answer: for each question
The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity and one focus at the Sun.The length of its major axis is 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 and the aphelion distance is Find theAnswer in AU and round to the nearest hundredth.

A)
B)
C)
D)
E)

Question 78

Select the correct Answer: for each question
Find the area of the region enclosed by one loop of the curve.

A)
B)
C)
D)
E)

Question 79

Select the correct Answer: for each question
Find parametric equations to represent the line segment from

A)
B)
C)
D)
E)

Question 80

Select the correct Answer: for each question
Find the surface area generated by rotating the lemniscate about the line

A)
B)
C)
D)
E)