Stewart – Calculus ET 8e Chapter 10 Form C
____ 19. 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 V be the origin. Find the
diameter of the opening |CD| , 19 cm from the vertex.
a.
b.
c. The equation is
d.
e. The equation is
f. The equation is
Stewart – Calculus ET 8e Chapter 10 Form C
____ 20. Find the vertices, foci and asymptotes of the hyperbola.
a. The foci are
b. The vertices are (4
5
, 5)
c. The foci are
d. The vertices are
e. The asymptote is
f. The asymptotes are
Stewart – Calculus ET 8e Chapter 10 Form C
Answer Key
Stewart – Calculus ET 8e Chapter 10 Form D
Select the correct answer for each question.
____ 1. Find parametric equations to represent the line segment from .
a.
b.
c.
d.
e.
____ 2. The curve cross itself at some point . Find the
equations of both tangent lines at that point.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 3. Find the surface area generated by rotating the lemniscate about the line .
a.
b.
c.
d.
e.
____ 4. 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 , 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 L = mi due east of station A on a coastline. A
ship received the signal from B microseconds (µs) before it received the signal from A.
Assuming that radio signals travel at a speed of ft /µs and if the ship is due north of B, how
far off the coastline is the ship? Round your answer to the nearest mile.
a. miles
b. miles
c. miles
d. miles
e. miles
Stewart – Calculus ET 8e Chapter 10 Form D
____ 5. 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
2.3 km, and its aphelion distance (maximum distance from the planet to the sun) is
approximately 2.7 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
____ 6. 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.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 7. Describe the motion of a particle with position as t 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
.
____ 8. 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
Stewart – Calculus ET 8e Chapter 10 Form D
____ 9. 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
____ 10. Find a polar equation for the curve represented by the given Cartesian equation.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 11. Find the polar equation for the curve represented by the given Cartesian equation.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 12. Sketch the polar curve with the given equation.
a.
1
c.
123456
b.
123456
d.
1
Stewart – Calculus ET 8e Chapter 10 Form D
____ 13. Find the length of the polar curve.
a.
b.
c.
d.
e. None of these
____ 14. Find the area of the region enclosed by one loop of the curve.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 15. Find the area of the region that lies inside the first curve and outside the second curve.
a.
b.
c.
d.
e.
____ 16. Find an equation of the parabola with focus and directrix .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 17. Find an equation for the conic that satisfies the given conditions.
ellipse, foci , length of major axis 8
a.
b.
c.
d.
e.
____ 18. Write a polar equation in r and
of an ellipse with the focus at the origin, with the eccentricity and
directrix .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form D
____ 19. 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 the answer in AU and round to the nearest hundredth.
a. AU
b. AU
c. AU
d. AU
e. AU
____ 20. 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.
Stewart – Calculus ET 8e Chapter 10 Form D
Answer Key
Stewart – Calculus ET 8e Chapter 10 Form E
____ 1. Find parametric equations to represent the line segment from .
Select the correct answer.
a.
b.
c.
d.
e.
2. Describe the motion of a particle with position as t varies in the given interval .
3. Find the point(s) on the curve where the tangent is horizontal.
4. Find an equation of the tangent to the curve at the point corresponding to the given value of the
parameter.
____ 5. Find a polar equation for the curve represented by the given Cartesian equation.
Select the correct answer.
a.
b.
c.
d.
e.
6. Find the slope of the tangent line to the given polar curve at the point specified by the value of .
7. Find the area of the region that lies inside the first curve and outside the second curve.
Stewart – Calculus ET 8e Chapter 10 Form E
8. Graph of the following curve is given. Find its length.
123456 r
9. The graph of the following curve is given. Find the area that it encloses.
4 8 12 16 20 24
Stewart – Calculus ET 8e Chapter 10 Form E
____ 10. Find an equation for the conic that satisfies the given conditions. Select the correct answer.
parabola, vertex (0, 0), focus (0, – )
a.
b.
c.
d.
e.
11. Find an equation for the conic that satisfies the given conditions.
hyperbola, foci (0, ± ) , vertices (0, ± )
12. Find an equation of the hyperbola centered at the origin that satisfies the given condition.
Vertices: (± 4, 0), asymptotes: y = ±
7
4
x
13. Find an equation of the hyperbola with vertices and asymptotes .
14. Find an equation for the conic that satisfies the given conditions.
ellipse, foci , length of major axis 8
Stewart – Calculus ET 8e Chapter 10 Form E
____ 15. 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 , 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 L = mi due east of station A on a coastline. A
ship received the signal from B microseconds (µs) before it received the signal from A.
Assuming that radio signals travel at a speed of ft /µs 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.
a. miles
b. miles
c. miles
d. miles
e. miles
16. Find an equation of the ellipse that satisfies the given conditions.
Foci: (0, ± 1), vertices (0, ± 6)
17. 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 the answer in AU and round to the nearest hundredth.
18. Write a polar equation of the conic that has a focus at the origin, eccentricity
7
2
, and directrix
. Identify the conic.
Stewart – Calculus ET 8e Chapter 10 Form E
____ 19. Find the equation of the directrix of the conic. Select the correct answer.
a.
b.
c.
d.
e.
20. 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.