Stewart – Calculus ET 8e Chapter 10 Form E
Answer Key
Stewart – Calculus ET 8e Chapter 10 Form F
1. Find parametric equations to represent the line segment from .
Select the correct answer.
a.
b.
c.
d.
e.
2. 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?
3. Describe the motion of a particle with position as t varies in the given interval .
4. Find the point(s) on the curve where the tangent is horizontal. Select the correct answer.
a.
b.
c.
d.
e. None of these
Stewart – Calculus ET 8e Chapter 10 Form F
5. Find the length of the curve. Select the correct answer.
, ,
a.
b.
c.
d.
e. None of these
6. Find the exact area of the surface obtained by rotating the given curve about the x-axis.
7. Find the point(s) of intersection of the curves and .
8. Find the surface area generated by rotating the lemniscate about the line .
9. Find the area of the region enclosed by one loop of the curve. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form F
10. Graph of the following curve is given. Find its length. Select the correct answer.
123456 r
a. L 25
b. L 24
c. L 26
d. L 32
e. L 20
11. Find an equation for the conic that satisfies the given conditions.
parabola, vertex (0, 0), focus (0, – )
12. Find an equation of the parabola with focus and directrix .
13. Find an equation of the hyperbola with vertices and asymptotes .
Stewart – Calculus ET 8e Chapter 10 Form F
14. 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
15. Match the equation with the correct graph.
16. Find an equation of the ellipse that satisfies the given conditions.
Foci: (0, ± 8), vertices (0, ± 9)
17. Write a polar equation in r and
of an ellipse with the focus at the origin, with the eccentricity and
directrix .
Stewart – Calculus ET 8e Chapter 10 Form F
18. Write a polar equation in r and of an ellipse with the focus at the origin, with the eccentricity and vertex
at .
19. 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.
Select the correct answer.
a. km
b. km
c. km
d. km
e. km
20. Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and
directrix .
Stewart – Calculus ET 8e Chapter 10 Form F
Answer Key
Stewart – Calculus ET 8e Chapter 10 Form F
Stewart – Calculus ET 8e Chapter 10 Form G
1. Find parametric equations to represent the line segment from .
____ 2. Find the point(s) on the curve where the tangent is horizontal. Select the correct answer.
a.
b.
c.
d.
e. None of these
3. Find the exact area of the surface obtained by rotating the given curve about the x-axis.
4. Find a polar equation for the curve represented by the given Cartesian equation.
5. Find the point(s) of intersection of the curves and .
6. Find the surface area generated by rotating the lemniscate about the line .
____ 7. Find the area of the region that lies inside the first curve and outside the second curve.
Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form G
8. The graph of the following curve is given. Find the area that it encloses.
4 8 12 16 20 24
9. Find an equation of the hyperbola centered at the origin that satisfies the given condition.
Vertices: (± 4, 0), asymptotes: y = ±
7
4
x
10. Find an equation of the hyperbola with vertices and asymptotes .
____ 11. Find an equation for the conic that satisfies the given conditions. Select the correct answer.
ellipse, foci , length of major axis 8
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form G
____ 12. 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
13. Match the equation with the correct graph.
14. Find an equation of the ellipse that satisfies the given conditions.
Foci: (0, ± 1), vertices (0, ± 6)
15. Find an equation of the ellipse that satisfies the given conditions.
Foci: (0, ± 8), vertices (0, ± 9)
16. Write a polar equation in r and
of an ellipse with the focus at the origin, with the eccentricity and
directrix .
Stewart – Calculus ET 8e Chapter 10 Form G
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. Find the equation of the directrix of the conic.
19. Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and
directrix .
____ 20. Find the eccentricity of the conic. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form G
Answer Key
Stewart – Calculus ET 8e Chapter 10 Form G
Stewart – Calculus ET 8e Chapter 10 Form H
1. Find parametric equations to represent the line segment from .
2. Find the point(s) on the curve where the tangent is horizontal.
____ 3. Find the length of the curve. Select the correct answer.
, ,
a.
b.
c.
d.
e. None of these
4. Find an equation of the tangent to the curve at the point corresponding to the given value of the
parameter.
5. Find the exact area of the surface obtained by rotating the given curve about the x-axis.
6. The curve cross itself at some point . Find the
equations of both tangent lines at that point.
Stewart – Calculus ET 8e Chapter 10 Form H
____ 7. Find the polar equation for the curve represented by the given Cartesian equation.
Select the correct answer.
a.
b.
c.
d.
e.
8. Find the length of the polar curve.
9. Find the surface area generated by rotating the lemniscate about the line .
10. Find the area of the region that lies inside the first curve and outside the second curve.
Stewart – Calculus ET 8e Chapter 10 Form H
____ 11. Graph of the following curve is given. Find its length. Select the correct answer.
123456 r
a. L 25
b. L 24
c. L 26
d. L 32
e. L 20
12. Find an equation of the parabola with focus and directrix .
13. Find an equation of the hyperbola with vertices and asymptotes .
14. Write a polar equation in r and
of an ellipse with the focus at the origin, with the eccentricity and
directrix .
15. 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.
Stewart – Calculus ET 8e Chapter 10 Form H
16. Write a polar equation of the conic that has a focus at the origin, eccentricity
7
2
, and directrix
. Identify the conic.
17. 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.
____ 18. 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.
Select the correct answer.
a. km
b. km
c. km
d. km
e. km
19. Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and
directrix .
____ 20. Find the eccentricity of the conic. Select the correct answer.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 10 Form H
Answer Key