Stewart – Calculus ET 8e Chapter 12 Form A
1.
a. Find an equation of the sphere that passes through the point and has center .
b. Find the curve in which this sphere intersects the xy-plane.
2. Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to
the xz-plane and units to the right of it.
3. Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes , , and .
4. Find an equation of the sphere that passes through the point and has center .
5. Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to
the coordinate planes. Find the length of the diagonal of the box.
6. Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.
7. Find the length of the median of side AB of the triangle with vertices
8. A woman walks due west on the deck of a ship at mi/h. The ship is moving north at a speed of
mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the
nearest tenth.
9. The tension T at each end of the chain has magnitude N and makes an angle with the
horizontal. What is the weight of the chain? Round the result to the nearest hundredth.
10. Velocities have both direction and magnitude and thus are vectors. The magnitude of a velocity
vector is called speed. Suppose that a wind is blowing from the direction N W at a speed of
km/h. (This means that the direction from which the wind blows is west of the northerly
direction.) A pilot is steering a plane in the direction N E at an airspeed (speed in still air) of
km/h. The true course, or track, of the plane is the direction of the resultant of the velocity
vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant.
Find the ground speed of the plane. Round the result to the nearest hundredth.
Stewart – Calculus ET 8e Chapter 12 Form A
11.
Calculate the given quantities if
12. A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is cm
long.
Find the magnitude of the torque about P correct to two decimal places.
Let , , .
13. Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.
14. Find the equation of the line through and perpendicular to the plane
15. Find parametric equations for the line through and
16. Find an equation of the plane that passes through the line of intersection of the planes and
is perpendicular to the plane
17. Find parametric equations for the line through and parallel to the vector
18. Find the point of intersection.
Stewart – Calculus ET 8e Chapter 12 Form A
19. Find an equation of the plane through the origin and parallel to the plane
20. Find an equation for the surface obtained by rotating the line about the x-axis.
Stewart – Calculus ET 8e Chapter 12 Form A
Answer Key
Stewart – Calculus ET 8e Chapter 12 Form B
1.
a. Find an equation of the sphere that passes through the point and has center .
b. Find the curve in which this sphere intersects the xy-plane.
2. Write an inequality to describe the half-space consisting of all points to the left of a plane parallel to
the xz-plane and units to the right of it.
3. Write inequalities to describe the solid rectangular box in the first octant bounded by the
planes , , and .
4. Write an inequality to describe the region consisting of all points between (but not on) the
spheres of radius and centered at the origin.
5. Find an equation of the sphere that passes through the point and has center .
6. Draw a rectangular box with the origin and as opposite vertices and with its faces parallel to
the coordinate planes. Find the length of the diagonal of the box.
7. Write inequalities to describe the solid upper hemisphere of the sphere of radius centered at the origin.
8. Find the length of the median of side AB of the triangle with vertices
9. A woman walks due west on the deck of a ship at mi/h. The ship is moving north at a speed of
mi/h. Find the speed of the woman relative to the surface of the water. Round the result to the
nearest tenth.
10. Calculate the given quantities if
Stewart – Calculus ET 8e Chapter 12 Form B
11. A bicycle pedal is pushed by a foot with a force of a-N as shown. The shaft of the pedal is cm
long.
Find the magnitude of the torque about P correct to two decimal places.
Let , , .
12. Find the magnitude of the torque about P correct to two decimal places if a d-lb force is applied as
shown. Let ft, ft, , lb.
13. Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.
14. Find the equation of the line through and perpendicular to the plane
15. Find parametric equations for the line through and
16. Find an equation of the plane that passes through the line of intersection of the planes and
is perpendicular to the plane
17. Find parametric equations for the line through and parallel to the vector
Stewart – Calculus ET 8e Chapter 12 Form B
18. Find the point of intersection.
19. Find an equation of the plane through the origin and parallel to the plane
20. Find an equation for the surface obtained by rotating the line about the x-axis.
Stewart – Calculus ET 8e Chapter 12 Form B
Answer Key
Stewart – Calculus ET 8e Chapter 12 Form C
Select the correct answer for each question.
____ 1. Suppose you start at the origin, move along the x-axis a distance of units in the positive direction,
and then move downward a distance of units. What are the coordinates of your position?
a.
b.
c.
d.
e.
____ 2. Find the center and radius of the sphere.
a.
b.
c.
d.
e. none of these
____ 3. Find the standard equation of the sphere with center C and radius r.
C (4, –1, 5); r = 8
a. (x + 4)2 + (y – 1)2 + (z + 5)2 = 64
b. (x + 4)2 + (y – 1)2 + (z + 5)2 = 8
c. (x – 4)2 + (y + 1)2 + (z – 5)2 = 8
d. (x – 4)2 + (y + 1)2 + (z – 5)2 = 64
____ 4. Find the distance from to the xy-planes.
a. 8
b. 16
c. 10
d. 2
e. 4
Stewart – Calculus ET 8e Chapter 12 Form C
____ 5. Find an equation of the sphere with center that touches the xy-plane.
a.
b.
c.
d.
e.
____ 6. Determine whether the given points are collinear.
A (2, 3, 2), B (–4, 0, 5), and C (4, 4, 1)
a. Not collinear b. Collinear
____ 7. Find the midpoint of the line segment joining the given points.
(4, –2, –3) and (6, –4, –1)
a. (
5
,
3
,
2
) c. (
1
,
3
,
1
)
b. (
1
,
1
,
1
) d. (
1
,
1
,
2
)
____ 8. Find the center and the radius of the sphere that has the given equation.
+ + – 4x + 8y = 0
a. ( 2, –4, 0),
25
c. ( –2, 4, 0),
25
b. ( –2, 4, 0), 20 d. ( 2, –4, 0), 20
Stewart – Calculus ET 8e Chapter 12 Form C
____ 9. Plot the given points in a three-dimensional coordinate system.
a. c.
b. d.
Stewart – Calculus ET 8e Chapter 12 Form C
____ 10. Find the unit vectors that are parallel to the tangent line to the curve at the point .
a.
b.
c.
d.
e.
____ 11. Find the work done by a force that moves an object from the point to the
point along a straight line. The distance is measured in meters and the force in newtons.
a.
b.
c.
d.
e.
____ 12. Find the cross product .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 12 Form C
____ 13. Let and let u be a vector with length that starts at the origin and rotates in the xy – plane.
Find the maximum value of the length of the vector .
a.
b.
c.
d.
e.
____ 14. Find a nonzero vector orthogonal to the plane through the points P, Q, and R.
a.
b.
c.
d.
e. None of these
____ 15. Find correct to three decimal places where , , .
a.
b.
c.
d.
e.
____ 16. Find the distance between the planes.
a.
b.
1
3
c.
d.
e.
Stewart – Calculus ET 8e Chapter 12 Form C
____ 17. Find, correct to the nearest degree, the angle between the planes.
a.
b.
c.
d.
e. none of these
____ 18. Find parametric equations for the line through the point that is parallel to the plane
and perpendicular to the line
a.
b.
c.
d.
e.
____ 19. Find the distance between the point and the plane Round your answer to
two decimal place.
a.
b.
c.
d.
e.
____ 20. Find an equation for the surface obtained by rotating the parabola about the y-axis.
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 12 Form C
Answer Key
Stewart – Calculus ET 8e Chapter 12 Form D
Select the correct answer for each question.
____ 1. Suppose you start at the origin, move along the x-axis a distance of units in the positive direction,
and then move downward a distance of units. What are the coordinates of your position?
a.
b.
c.
d.
e.
____ 2. Find that the midpoint of the line segment from to .
a.
b.
c.
d.
e.
____ 3. Find the length of each side of the triangle ABC and determine whether the triangle is an isosceles triangle, a
right triangle, both, or neither.
A (–1, 0, 1), B (1, 1, –1), C (1, 1, 1)
a. 3, 2, , right
b. 1, 2, , neither
c. 1, 1, , both
d. 1, 1, , isosceles
____ 4. Find the distance from to the xy-planes.
a. 8
b. 16
c. 10
d. 2
e. 4
Stewart – Calculus ET 8e Chapter 12 Form D
____ 5. Find an equation of the sphere with center that touches the xy-plane.
a.
b.
c.
d.
e.
____ 6. Sketch the plane in a three-dimensional space represented by the equation.
z = –1
a. c.
b. d.
Stewart – Calculus ET 8e Chapter 12 Form D
____ 7. Determine whether the given points are collinear.
A (2, 3, 2), B (–4, 0, 5), and C (4, 4, 1)
a. Not collinear b. Collinear
____ 8. Find the midpoint of the line segment joining the given points.
(4, –2, –3) and (6, –4, –1)
a. (
5
,
3
,
2
) c. (
1
,
3
,
1
)
b. (
1
,
1
,
1
) d. (
1
,
1
,
2
)
____ 9. Find the center and the radius of the sphere that has the given equation.
+ + – 4x + 8y = 0
a. ( 2, –4, 0),
25
c. ( –2, 4, 0),
25
b. ( –2, 4, 0), 20 d. ( 2, –4, 0), 20
____ 10. Find the work done by a force that moves an object from the point to the
point along a straight line. The distance is measured in meters and the force in newtons.
a.
b.
c.
d.
e.
____ 11. Find the cross product .
a.
b.
c.
d.
e.
Stewart – Calculus ET 8e Chapter 12 Form D
____ 12. Let and let u be a vector with length that starts at the origin and rotates in the xy – plane.
Find the maximum value of the length of the vector .
a.
b.
c.
d.
e.
____ 13. Find a nonzero vector orthogonal to the plane through the points P, Q, and R.
a.
b.
c.
d.
e. None of these
____ 14. Find an equation of the set of all points equidistant from the points and
a.
b.
c.
d.
e.
____ 15. Find the distance between the planes.
a.
b.
1
3
c.
d.
e.
Stewart – Calculus ET 8e Chapter 12 Form D
____ 16. Find, correct to the nearest degree, the angle between the planes.
a.
b.
c.
d.
e. none of these
____ 17. Which of the given lines is parallel to the line
a.
b.
c.
d.
e.
____ 18. Find the distance between the point and the plane Round your answer to
two decimal place.
a.
b.
c.
d.
e.
____ 19. Find an equation for the surface consisting of all points P for which the distance from P to the x-axis
is times the distance from P to the yz-plane.
a.
b.
c.
d.
e.