27.
The orbit of Halley’s comet, last seen in 1986 and due to
return in 2062, is an ellipse with eccentricity 0.97 and one
focus at the sun. The length of its major axis is 36.18 AU.
[An astronomical unit (AU) is the mean distance between the
earth and the sun, about 93 million miles.] Find a polar equa-
tion for the orbit of Halley’s comet. What is the maximum
distance from the comet to the sun?
28.
The Hale-Bopp comet, discovered in 1995, has an elliptical
orbit with eccentricity 0.9951 and the length of the major
axis is 356.5 AU. Find a polar equation for the orbit of this
comet. How close to the sun does it come?
29.
The planet Mercury travels in an elliptical orbit with eccen-
tricity
. Its minimum distance from the sun is
km. Find its maximum distance from the sun.
30.
The distance from the planet Pluto to the sun is
km at perihelion and
km at aphelion.
Find the eccentricity of Pluto’s orbit.
31.
Using the data from Exercise 29, find the distance traveled by
the planet Mercury during one complete orbit around the sun.
(If your calculator or computer algebra system evaluates defi-
nite integrals, use it. Otherwise, use Simpson’s Rule.)
e
y
d
r
ed
1
e
sin
e
y
d
r
ed
1
e
sin
r
c
1
cos
r
d
1
cos
0.093
2.28
10
8
km
0.048
1.56
10
9
km
0.206
4.6
10
7
4.43
10
9
7.37
10
9
10.6
Exercises
;
Graphing calculator or computer required
1.
Homework Hints available at stewartcalculus.com
© Dean Ketelsen

CHAPTER 10
REVIEW
709
10
Review
1.
(a)
What is a parametric curve?
(b) How do you sketch a parametric curve?
2.
(a)
How do you find the slope of a tangent to a parametric
curve?
(b) How do you find the area under a parametric curve?
3.
Write an expression for each of the following:
(a)
The length of a parametric curve
(b) The area of the surface obtained by rotating a parametric
curve about the
4.
(a)
Use a diagram to explain the meaning of the polar coordi-
nates
of a point.
(b) Write equations that express the Cartesian coordinates
of a point in terms of the polar coordinates.
(c)
What equations would you use to find the polar coordinates
of a point if you knew the Cartesian coordinates?
5.
(a)
How do you find the slope of a tangent line to a polar
curve?
(b) How do you find the area of a region bounded by a polar
curve?
(c)
How do you find the length of a polar curve?
x
-axis
r
,
x
,
y
6.
(a)
Give a geometric definition of a parabola.
(b) Write an equation of a parabola with focus
and direc-
trix
. What if the focus is
and the directrix
is
?
7.
(a)
Give a definition of an ellipse in terms of foci.
(b) Write an equation for the ellipse with foci
and
vertices
.
8.
(a)
Give a definition of a hyperbola in terms of foci.
(b) Write an equation for the hyperbola with foci
and
vertices
.
(c)
Write equations for the asymptotes of the hyperbola in
part (b).
9.
(a)
What is the eccentricity of a conic section?
(b) What can you say about the eccentricity if the conic section
is an ellipse? A hyperbola? A parabola?