11.5 Lines and Planes in Space
712
11.5 Lines and Planes in Space
Answer Section
713
Chapter 11: Vectors and the Geometry of Space
21.
11.6 Surfaces in Space
714
11.6 Surfaces in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Identify the equation of the following graph.
a.
b.
c.
d.
e.
715 Chapter 11: Vectors and the Geometry of Space
____ 2. Identify the equation of the following graph.
a.
b.
c.
d.
e.
____ 3. Identify the following quadric surface.
hyperboloid of one sheet
elliptic cone
hyperboloid of two sheets
elliptic paraboloid
ellipsoid
11.6 Surfaces in Space
716
____ 4. Identify the following quadric surface.
hyperboloid of two sheets
elliptic cone
ellipsoid
elliptic paraboloid
hyperboloid of one sheet
____ 5. Identify the following quadric surface.
ellipsoid
elliptic cone
elliptic paraboloid
hyperboloid of one sheet
hyperboloid of two sheets
____ 6. Identify the following quadric surface.
elliptic paraboloid
elliptic cone
hyperboloid of two sheets
hyperboloid of one sheet
ellipsoid
____ 7. Identify the following quadric surface.
hyperboloid of one sheet
elliptic cone
elliptic paraboloid
hyperboloid of two sheets
hyperbolic paraboloid
717 Chapter 11: Vectors and the Geometry of Space
____ 8. Identify the following quadric surface.
Elliptic cone
Hyperboloid of two sheets
Elliptic paraboloid
ellipsoid
Hyperboloid of one sheet
____ 9.
Find an equation of the surface of revolution generated by revolving the curve given
below in the indicated coordinate plane about the given axis.
Equation of Curve
Coordinate Plane
Axis of Revolution
a.
b.
c.
d.
e.
____ 10. Find an equation for the surface of revolution generated by revolving the curve
in the xz–plane about the z–axis.
a.
b.
c.
d.
e.
11.6 Surfaces in Space
718
____ 11.
Given the equation of a surface of revolution
find the equation of
its generating curve in the yz–plane rotating about the y–axis.
a.
b.
c.
d.
e.
____
12.
Use the shell method to find the volume of the solid below the surface formed
by revolving the curve
about the z-axis and above the xy-plane.
a.
b.
c.
d.
e.
____
13.
Find the length of the minor axis of the ellipse generated when the surface
is intersected by the plane
.
10 10
5 5
d.
e.
____ 14. Find the equation of the surface satisfying the conditions, and identify the surface.
The set of points equidistant from the point (2, 4, 5 ) and the plane .
a.
b.
c.
d.
e.
719 Chapter 11: Vectors and the Geometry of Space
____ 15. Some planet is an oblate ellipsoid rather than a sphere because of the forces caused
by its rotation. The equatorial radius of this planet is 4961 miles and the polar radius is
4687 miles. Find the equation of the ellipsoid. (Assume that the center of a planet is at the
origin and that the trace formed by the plane
corresponds to the equator.)
a.
b.
c.
d.
e.
____
16.
The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve
in the yz-plane about
the z-axis. Find an equation for the surface of revolution.
a.
b.
c.
d.
e.
____
17.
The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve
in the yz-plane about
the z-axis. All measurements are in centimeters and the bushing is set on the xy-plane. Use the shell
method to find its volume. Round your answer to one decimal place.
a.
b.
c.
d.
e.
11.6 Surfaces in Space
720
____ 18.
The top of a rubber bushing designed to absorb vibrations in an automobile is the
surface of revolution generated by revolving the curve
in the yz-plane
about the z-axis. The bushing has a hole of 4 centimeters in diameter through its center and parallel to
the axis of revolution. All measurements are in centimeters and the bushing is set on the xy-plane.
Find the volume of the rubber bushing. Round your answer to two decimal places.
a.
b.
c.
d.
e.
721 Chapter 11: Vectors and the Geometry of Space
11.6 Surfaces in Space
Answer Section
11.7 Cylindrical and Spherical Coordinates
722
11.7 Cylindrical and Spherical Coordinates
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Convert the following point from cylindrical coordinates to rectangular coordinates.
a.
b.
c.
d.
e.
____ 2. Convert the following point from rectangular coordinates to cylindrical coordinates. Give
any angles in radians.
a.
b.
c.
d.
e.
____ 3. Find an equation in cylindrical coordinates for the equation given in rectangular
coordinates.
a.
b.
c.
723 Chapter 11: Vectors and the Geometry of Space
d.
e.
____ 4. Find an equation in cylindrical coordinates for the equation given in rectangular
coordinates.
a.
b.
c.
d.
e.
____ 5. Find an equation in rectangular coordinates for the equation given in cylindrical
coordinates.
a.
b.
c.
d.
e.
11.7 Cylindrical and Spherical Coordinates
724
____
6.
Find an equation in rectangular coordinates for the equation given in cylindrical
coordinates.
a.
b.
c.
d.
e.
____
7.
Find an equation in rectangular coordinates for the equation given in cylindrical
coordinates.
a.
b.
c.
d.
e.
____
8.
Convert the point
from rectangular coordinates to spherical
coordinates.
a.
b.
725 Chapter 11: Vectors and the Geometry of Space
c.
d.
e.
____ 9. Convert the point from spherical coordinates to rectangular coordinates.
a.
b.
c.
d.
e.
____ 10. Find an equation in spherical coordinates for the equation given in rectangular
coordinates.
a.
b.
c.
d.
e.
11.7 Cylindrical and Spherical Coordinates
726
____ 11.
Find an equation in spherical coordinates for the equation given in rectangular
coordinates.
a.
b.
c.
d.
e.
____ 12. Find an equation in rectangular coordinates for the equation given in spherical
coordinates. Express the coefficients to two decimal places.
a.
b.
c.
d.
e.
____ 13. Find an equation in rectangular coordinates for the equation given in spherical
coordinates.
a.
b.
c.
727 Chapter 11: Vectors and the Geometry of Space
d.
e.
____ 14. Convert the following point from spherical coordinates to cylindrical coordinates.
a.
b.
c.
d.
e.
____ 15. Identify the equation (written in terms of cylindrical or spherical coordinates) of the
following graph.
11.7 Cylindrical and Spherical Coordinates
728
a.
b.
c.
d.
e.
____ 16. Identify the equation (written in terms of cylindrical or spherical coordinates) of the
following graph.
a.
b.
c.
d.
e.
729
Chapter 11: Vectors and the Geometry of Space
____
17.
Graph the solid described in spherical coordinates.
a. d.
b. e.
c.
11.7 Cylindrical and Spherical Coordinates
730
11.7 Cylindrical and Spherical Coordinates
Answer Section