تكلم مع الذكاء الاصطناعي
Deck 11: Vectors and the Geometry of Space
ملء الشاشة (f)
سؤال
Identify the following quadric surface. 
A) hyperboloid of two sheets
B) elliptic cone
C) ellipsoid
D) elliptic paraboloid
E) hyperboloid of one sheet
A) hyperboloid of two sheets
B) elliptic cone
C) ellipsoid
D) elliptic paraboloid
E) hyperboloid of one sheet
سؤال
Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places. Initial point: (3, -3, -4)
Terminal point: (8, -7, -1)
A)
or 7.2801
B)
or 7.6158
C)
or 7.0711
D)
or 4.3589
E)
or 7.8102
Terminal point: (8, -7, -1)
A)
or 7.2801B)
or 7.6158C)
or 7.0711D)
or 4.3589E)
or 7.8102 سؤال
Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find an equation in cylindrical coordinates for the equation given in rectangular coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the magnitude of the vector given below. Round your answer to four decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the vector v with the magnitude 
and direction of 
.
A)
B)
C)
D)
E)
and direction of .
A)
B)
C)
D)
E)
سؤال
Complete the square to write the following equation in the standard form for a sphere. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
A chute at the top of a grain elevator of a combine funnels the grain into a bin (see figure). Find the angle between two adjacent sides. Round your answer to one decimal place. 
A) 83.7
B) 91.6
C) 90.7
D) 86.4
E) 181.6
A) 83.7
B) 91.6
C) 90.7
D) 86.4
E) 181.6
سؤال
Find the distance between the points given below.
(3, 1, 4 ), ( 6, 6, 7 )
A)
or 6.5574 to four decimal places.
B)
or 6.4807 to four decimal places.
C)
or 6.6332 to four decimal places.
D)
or 6.0828 to four decimal places.
E)
or 6.7082 to four decimal places.
(3, 1, 4 ), ( 6, 6, 7 )
A)
or 6.5574 to four decimal places.B)
or 6.4807 to four decimal places.C)
or 6.6332 to four decimal places.D)
or 6.0828 to four decimal places.E)
or 6.7082 to four decimal places. سؤال
Identify the equation of the following graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Sketch the graph of the cross product of the unit vector 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Find the standard equation of the sphere with center (4, 1, 1), and radius 2.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Graph the solid described in spherical coordinates. 
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
سؤال
Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither. 
A) Neither parallel nor orthogonal
B) Parallel, same direction
C) Orthogonal
D) Parallel, opposite direction
A) Neither parallel nor orthogonal
B) Parallel, same direction
C) Orthogonal
D) Parallel, opposite direction
سؤال
Find the component form of the vector u with the given initial and terminal points. Initial point: (2, 1, 1 )
Terminal point: (7 , 10 , 3 )
A)
B)
C)
D)
E)
Terminal point: (7 , 10 , 3 )
A)
B)
C)
D)
E)
سؤال
Find a set of parametric equations of the line that passes through the point 
and is parallel to the xy-plane and yz-plane.
A)
B)
C)
D)
E)
and is parallel to the xy-plane and yz-plane.A)
B)
C)
D)
E)
سؤال
Find the standard equation of the sphere with center at 
and tangent to the yz-plane.
A)
B)
C)
D)
E)
and tangent to the yz-plane.A)
B)
C)
D)
E)
سؤال
Identify the equation of the following graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
The vector v and its initial point is given. Find the terminal point. 
, initial point 
A)
B)
C)
D)
E)
, initial point A)
B)
C)
D)
E)
سؤال
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 1 centimeter 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)
in the yz-plane about the z-axis. The bushing has a hole of 1 centimeter 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)
سؤال
To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a 
angle away from the vertical and the other makes a 
angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places. 
A)
B)
C)
D)
E)
angle away from the vertical and the other makes a angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places. A)
B)
C)
D)
E)
سؤال
Find a unit vector that is orthogonal to both 
and 
.
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
سؤال
Find the distance between the point 
and the plane 
.
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
سؤال
Find the triple scalar product 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Suppose a car is towed using a force of 1300 newtons. The chain used to pull the car makes a 
angle with the horizontal. Find the work done in towing the car 3 kilometers. Round your answer to two decimal places.
A) 7,070.20 km-N
B) 3,534.60 km-N
C) 3,549.60 km-N
D) 7,068.20 km-N
E) 1,648.21 km-N
angle with the horizontal. Find the work done in towing the car 3 kilometers. Round your answer to two decimal places.A) 7,070.20 km-N
B) 3,534.60 km-N
C) 3,549.60 km-N
D) 7,068.20 km-N
E) 1,648.21 km-N
سؤال
Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Given 
and 
, find 
.
A)
B)
C)
D)
E)
and , find . A)
B)
C)
D)
E)
سؤال
Three forces with magnitudes 
pounds, 
pounds and 
pounds act on an object at angles 
, 
, and 
respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)
A)
pounds at an angle of 
with the positive x-axis
B)
pounds at an angle of 
with the positive x-axis
C)
pounds at an angle of 
with the positive x-axis
D)
pounds at an angle of 
with the positive x-axis
E)
pounds at an angle of 
with the positive x-axis
pounds, pounds and pounds act on an object at angles , , and respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)A)
pounds at an angle of with the positive x-axisB)
pounds at an angle of with the positive x-axisC)
pounds at an angle of with the positive x-axisD)
pounds at an angle of with the positive x-axisE)
pounds at an angle of with the positive x-axis سؤال
Identify the following quadric surface. 
A) hyperboloid of one sheet
B) elliptic cone
C) elliptic paraboloid
D) hyperboloid of two sheets
E) hyperbolic paraboloid
A) hyperboloid of one sheet
B) elliptic cone
C) elliptic paraboloid
D) hyperboloid of two sheets
E) hyperbolic paraboloid
سؤال
Find an equation of a plane passing through the following three points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
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)
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)
سؤال
Find the set of parametric equations of the line through the point 
that is parallel to the line 
and 
.
A)
B)
C)
D)
E)
that is parallel to the line and . A)
B)
C)
D)
E)
سؤال
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)
in the yz-plane about the z-axis. Find an equation for the surface of revolution.A)
B)
C)
D)
E)
سؤال
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 4851 miles and the polar radius is 4660 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)
corresponds to the equator.)A)
B)
C)
D)
E)
سؤال
Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point 
where 
. Also, at time t, the second insect is at the point 
where 
. Assume distances are given in inches. Find the distance between the two insects at time 
. Round your answer to two decimal places.
A)
inches
B)
inches
C)
inches
D)
inches
E)
inches
where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.A)
inchesB)
inchesC)
inchesD)
inchesE)
inches سؤال
Identify the following quadric surface. 
A) hyperboloid of one sheet
B) elliptic cone
C) hyperboloid of two sheets
D) elliptic paraboloid
E) ellipsoid
A) hyperboloid of one sheet
B) elliptic cone
C) hyperboloid of two sheets
D) elliptic paraboloid
E) ellipsoid
سؤال
Suppose a plane is flying in the direction 
(relative to the west). Its speed with respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at 80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer to one decimal place. 
A)
B)
C)
D)
E)
(relative to the west). Its speed with respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at 80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer to one decimal place. A)
B)
C)
D)
E)
سؤال
Given 
, find 
.
A)
B)
C)
D)
E)
, find .A)
B)
C)
D)
E)
سؤال
Find vectors u and v whose initial and terminal points are given. Determine whether u and v are equivalent.
U: (4,2), (7,6) v: (-2,3), (1,7)
A)
, which are equivalent
B)
, which are not equivalent
C)
, which are equivalent
D)
, which are equivalent
E)
, which are equivalent
U: (4,2), (7,6) v: (-2,3), (1,7)
A)
, which are equivalentB)
, which are not equivalentC)
, which are equivalentD)
, which are equivalentE)
, which are equivalent سؤال
Convert the point 
from rectangular coordinates to spherical coordinates.
A)
B)
C)
D)
E)
from rectangular coordinates to spherical coordinates.A)
B)
C)
D)
E)
سؤال
Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point (-4, 3, 2 ) and the plane 
.
A)
, an elliptic paraboloid
B)
, a sphere
C)
, a hyperbolic paraboloid
D)
, a sphere
E)
, a hyperbolic paraboloid
.A)
, an elliptic paraboloidB)
, a sphereC)
, a hyperbolic paraboloidD)
, a sphereE)
, a hyperbolic paraboloid سؤال
An object is pulled 5 feet horizontally across a floor, using a force of 65 pounds. The direction of the force is 
above the horizontal. Find the work done. Round your answer to two decimal places.
A) 294.55 ft-lb
B) 322.14 ft-lb
C) 137.35 ft-lb
D) 5.62 ft-lb
E) 336.74 ft-lb
above the horizontal. Find the work done. Round your answer to two decimal places.A) 294.55 ft-lb
B) 322.14 ft-lb
C) 137.35 ft-lb
D) 5.62 ft-lb
E) 336.74 ft-lb
سؤال
Find a set of symmetric equations of the line through the points 
and 
.
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
سؤال
Convert the point from spherical coordinates to rectangular coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Suppose a plane is flying in the direction 
. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. 
A)
North of West
B)
North of West
C)
North of East
D)
North of West
E)
North of South
. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. A)
North of WestB)
North of WestC)
North of EastD)
North of WestE)
North of South سؤال
Find the area of the parallelogram with the vertices 
, and 
. Round your answer to two decimal places.
A) 122.00
B) 6.00
C) 11.05
D) 12.00
E) 3.46
, and . Round your answer to two decimal places.A) 122.00
B) 6.00
C) 11.05
D) 12.00
E) 3.46
سؤال
Find the direction cosines of the vector u given below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the volume of the parallelepiped with the following vertices. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Given 
and 
find 
.
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
سؤال
Suppose a plane flies at a constant groundspeed of 100 miles per hour due east and encounters a 40 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
سؤال
Find a set of symmetric equations of the line through the point 
parallel to the vector 
.
A)
B)
C)
D)
E)
parallel to the vector .A)
B)
C)
D)
E)
سؤال
A child applies the brakes on a bicycle by applying a downward force of 20 pounds on the pedal when the crank makes a 
angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. 
A) 15.32 ft-lb
B) 7.66 ft-lb
C) 91.93 ft-lb
D) 6.43 ft-lb
E) 12.86 ft-lb
angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. A) 15.32 ft-lb
B) 7.66 ft-lb
C) 91.93 ft-lb
D) 6.43 ft-lb
E) 12.86 ft-lb
سؤال
Convert the following point from cylindrical coordinates to rectangular coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Suppose a plane flies at a constant groundspeed of 300 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Determine whether the following planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection. 
A) The planes are neither parallel nor orthogonal, the angle of intersection is
.
B) The planes are parallel.
C) The planes are neither parallel nor orthogonal, the angle of intersection is
.
D) The planes are orthogonal.
A) The planes are neither parallel nor orthogonal, the angle of intersection is
.B) The planes are parallel.
C) The planes are neither parallel nor orthogonal, the angle of intersection is
.D) The planes are orthogonal.
سؤال
Suppose a toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a 
angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place. 
A) 1,200.0 ft-lb
B) 427.5 ft-lb
C) 2,350.2 ft-lb
D) 1,174.6 ft-lb
E) 2,348.2 ft-lb
angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place. A) 1,200.0 ft-lb
B) 427.5 ft-lb
C) 2,350.2 ft-lb
D) 1,174.6 ft-lb
E) 2,348.2 ft-lb
سؤال
Find the cross product of the unit vector 
.
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
سؤال
Find the direction cosines of the vector u given below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Given the equation of a surface of revolution 
, find the equation of its generating curve in the xy-plane rotating about the x-axis.
A)
B)
C)
D)
E)
, find the equation of its generating curve in the xy-plane rotating about the x-axis.A)
B)
C)
D)
E)
سؤال
Find a set of parametric equations of the line through the point 
parallel to the vector 
.
A)
B)
C)
D)
E)
parallel to the vector .A)
B)
C)
D)
E)
سؤال
Identify the following quadric surface. 
A) elliptic paraboloid
B) elliptic cone
C) hyperboloid of two sheets
D) hyperboloid of one sheet
E) ellipsoid
A) elliptic paraboloid
B) elliptic cone
C) hyperboloid of two sheets
D) hyperboloid of one sheet
E) ellipsoid
سؤال
Given 
, find 
.
A) -6
B) -24
C) -5
D) 4
E) 26
, find .A) -6
B) -24
C) -5
D) 4
E) 26
سؤال
Given 
, find 
.
A)
B)
C)
D)
E)
, find .A)
B)
C)
D)
E)
سؤال
Given 
and 
find 
.
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
سؤال
Given 
and 
find 
.
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
سؤال
Find the angle between the vectors for u and v given below. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find the distance between the point 
and the line given by the set of parametric equations 
.
A)
/3
B) 1
C)
/3
D) 3
E)
/1
and the line given by the set of parametric equations . A)
/3 B) 1
C)
/3 D) 3
E)
/1 سؤال
Given 
and 
, find 
.
A)
B)
C)
D)
E)
and , find .A)
B)
C)
D)
E)
سؤال
Find a set of parametric equations of the line through the points 
and 
.
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
سؤال
Suppose a 48,000-pound truck is parked on a 
slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place. 
A) 47,070.8 lb
B) 47,436.8 lb
C) 47,313.8 lb
D) 47,270.8 lb
E) 47,280.8 lb
slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place. A) 47,070.8 lb
B) 47,436.8 lb
C) 47,313.8 lb
D) 47,270.8 lb
E) 47,280.8 lb
سؤال
Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find an equation in spherical coordinates for the equation given in rectangular coordinates. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Given 
and 
, find 
.
A)
B)
C)
D)
E)
and , find .A)
B)
C)
D)
E)
سؤال
Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places. 
A) 2,771.28 ft-lb
B) 277.13 ft-lb
C) 226.27 ft-lb
D) 554.26 ft-lb
E) 160.00 ft-lb
A) 2,771.28 ft-lb
B) 277.13 ft-lb
C) 226.27 ft-lb
D) 554.26 ft-lb
E) 160.00 ft-lb
سؤال
To carry a 500-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a 
angle away from the vertical and the other makes a 
angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places.
A)
B)
C)
D)
E)
angle away from the vertical and the other makes a angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places.
A)
B)
C)
D)
E)
سؤال
Find the angle between the vectors for u and v. 
A) 46.88
B) 23.20
C) 66.80
D) 42.59
E) 88.85
A) 46.88
B) 23.20
C) 66.80
D) 42.59
E) 88.85
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ملء الشاشة (f)
Deck 11: Vectors and the Geometry of Space
1
Identify the following quadric surface.
A) hyperboloid of two sheets
B) elliptic cone
C) ellipsoid
D) elliptic paraboloid
E) hyperboloid of one sheet
A) hyperboloid of two sheets
B) elliptic cone
C) ellipsoid
D) elliptic paraboloid
E) hyperboloid of one sheet
E
2
Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places. Initial point: (3, -3, -4)
Terminal point: (8, -7, -1)
A) or 7.2801
B) or 7.6158
C) or 7.0711
D) or 4.3589
E) or 7.8102
Terminal point: (8, -7, -1)
A)
or 7.2801B)
or 7.6158C)
or 7.0711D)
or 4.3589E)
or 7.8102C
3
Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
C
4
Find an equation in cylindrical coordinates for the equation given in rectangular coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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5
Find the standard equation of a sphere that has diameter with the end points given below. (4, 8, 3), (2, 4, 7)
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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6
Find the magnitude of the vector given below. Round your answer to four decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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7
Find the vector v with the magnitude and direction of .
A)
B)
C)
D)
E)
and direction of .
A)
B)
C)
D)
E)
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8
Complete the square to write the following equation in the standard form for a sphere.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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9
A chute at the top of a grain elevator of a combine funnels the grain into a bin (see figure). Find the angle between two adjacent sides. Round your answer to one decimal place.
A) 83.7
B) 91.6
C) 90.7
D) 86.4
E) 181.6
A) 83.7
B) 91.6
C) 90.7
D) 86.4
E) 181.6
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10
Find the distance between the points given below.
(3, 1, 4 ), ( 6, 6, 7 )
A) or 6.5574 to four decimal places.
B) or 6.4807 to four decimal places.
C) or 6.6332 to four decimal places.
D) or 6.0828 to four decimal places.
E) or 6.7082 to four decimal places.
(3, 1, 4 ), ( 6, 6, 7 )
A)
or 6.5574 to four decimal places.B)
or 6.4807 to four decimal places.C)
or 6.6332 to four decimal places.D)
or 6.0828 to four decimal places.E)
or 6.7082 to four decimal places. فتح الحزمة
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11
Identify the equation of the following graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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12
Sketch the graph of the cross product of the unit vector .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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13
Find the standard equation of the sphere with center (4, 1, 1), and radius 2.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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14
Identify the equation (written in terms of cylindrical or spherical coordinates) of the following graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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15
Graph the solid described in spherical coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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16
Determine whether u and v are orthogonal, parallel and pointing in the same direction, parallel and pointing in opposite directions, or neither.
A) Neither parallel nor orthogonal
B) Parallel, same direction
C) Orthogonal
D) Parallel, opposite direction
A) Neither parallel nor orthogonal
B) Parallel, same direction
C) Orthogonal
D) Parallel, opposite direction
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17
Find the component form of the vector u with the given initial and terminal points. Initial point: (2, 1, 1 )
Terminal point: (7 , 10 , 3 )
A)
B)
C)
D)
E)
Terminal point: (7 , 10 , 3 )
A)
B)
C)
D)
E)
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18
Find a set of parametric equations of the line that passes through the point and is parallel to the xy-plane and yz-plane.
A)
B)
C)
D)
E)
and is parallel to the xy-plane and yz-plane.A)
B)
C)
D)
E)
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19
Find the standard equation of the sphere with center at and tangent to the yz-plane.
A)
B)
C)
D)
E)
and tangent to the yz-plane.A)
B)
C)
D)
E)
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20
Identify the equation of the following graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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21
The vector v and its initial point is given. Find the terminal point. , initial point
A)
B)
C)
D)
E)
, initial point A)
B)
C)
D)
E)
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22
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 1 centimeter 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)
in the yz-plane about the z-axis. The bushing has a hole of 1 centimeter 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)
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23
To carry a 100-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a angle away from the vertical and the other makes a angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places.
A)
B)
C)
D)
E)
angle away from the vertical and the other makes a angle as shown in the figure below. Find each rope's tension if the resultant force is vertical.Round your answer to two decimal places. A)
B)
C)
D)
E)
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24
Find a unit vector that is orthogonal to both and .
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
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25
Find the distance between the point and the plane .
A)
B)
C)
D)
E)
and the plane .A)
B)
C)
D)
E)
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26
Find the triple scalar product .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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27
Suppose a car is towed using a force of 1300 newtons. The chain used to pull the car makes a angle with the horizontal. Find the work done in towing the car 3 kilometers. Round your answer to two decimal places.
A) 7,070.20 km-N
B) 3,534.60 km-N
C) 3,549.60 km-N
D) 7,068.20 km-N
E) 1,648.21 km-N
angle with the horizontal. Find the work done in towing the car 3 kilometers. Round your answer to two decimal places.A) 7,070.20 km-N
B) 3,534.60 km-N
C) 3,549.60 km-N
D) 7,068.20 km-N
E) 1,648.21 km-N
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28
Find the area of a parallelogram that has the given vectors as adjacent sides. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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29
Given and , find .
A)
B)
C)
D)
E)
and , find . A)
B)
C)
D)
E)
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30
Three forces with magnitudes pounds, pounds and pounds act on an object at angles , , and respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)
A) pounds at an angle of with the positive x-axis
B) pounds at an angle of with the positive x-axis
C) pounds at an angle of with the positive x-axis
D) pounds at an angle of with the positive x-axis
E) pounds at an angle of with the positive x-axis
pounds, pounds and pounds act on an object at angles , , and respectively, with the positive x-axis. Find the direction and magnitude of the resultant force. (The choices below are given to two decimal places.)A)
pounds at an angle of with the positive x-axisB)
pounds at an angle of with the positive x-axisC)
pounds at an angle of with the positive x-axisD)
pounds at an angle of with the positive x-axisE)
pounds at an angle of with the positive x-axis فتح الحزمة
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31
Identify the following quadric surface.
A) hyperboloid of one sheet
B) elliptic cone
C) elliptic paraboloid
D) hyperboloid of two sheets
E) hyperbolic paraboloid
A) hyperboloid of one sheet
B) elliptic cone
C) elliptic paraboloid
D) hyperboloid of two sheets
E) hyperbolic paraboloid
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32
Find an equation of a plane passing through the following three points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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33
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)
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)
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34
Find the set of parametric equations of the line through the point that is parallel to the line and .
A)
B)
C)
D)
E)
that is parallel to the line and . A)
B)
C)
D)
E)
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35
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)
in the yz-plane about the z-axis. Find an equation for the surface of revolution.A)
B)
C)
D)
E)
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36
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 4851 miles and the polar radius is 4660 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)
corresponds to the equator.)A)
B)
C)
D)
E)
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37
Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.
A) inches
B) inches
C) inches
D) inches
E) inches
where . Also, at time t, the second insect is at the point where . Assume distances are given in inches. Find the distance between the two insects at time . Round your answer to two decimal places.A)
inchesB)
inchesC)
inchesD)
inchesE)
inches فتح الحزمة
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38
Identify the following quadric surface.
A) hyperboloid of one sheet
B) elliptic cone
C) hyperboloid of two sheets
D) elliptic paraboloid
E) ellipsoid
A) hyperboloid of one sheet
B) elliptic cone
C) hyperboloid of two sheets
D) elliptic paraboloid
E) ellipsoid
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39
Suppose a plane is flying in the direction (relative to the west). Its speed with respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at 80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer to one decimal place.
A)
B)
C)
D)
E)
(relative to the west). Its speed with respect to the air is 600 kilometers per hour. The wind at the plane's altitude is from the southwest at 80 kilometers per hour (see figure). What is its speed with respect to the ground? Round your answer to one decimal place. A)
B)
C)
D)
E)
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40
Given , find .
A)
B)
C)
D)
E)
, find .A)
B)
C)
D)
E)
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41
Find vectors u and v whose initial and terminal points are given. Determine whether u and v are equivalent.
U: (4,2), (7,6) v: (-2,3), (1,7)
A) , which are equivalent
B) , which are not equivalent
C) , which are equivalent
D) , which are equivalent
E) , which are equivalent
U: (4,2), (7,6) v: (-2,3), (1,7)
A)
, which are equivalentB)
, which are not equivalentC)
, which are equivalentD)
, which are equivalentE)
, which are equivalent فتح الحزمة
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42
Convert the point from rectangular coordinates to spherical coordinates.
A)
B)
C)
D)
E)
from rectangular coordinates to spherical coordinates.A)
B)
C)
D)
E)
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43
Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point (-4, 3, 2 ) and the plane .
A) , an elliptic paraboloid
B) , a sphere
C) , a hyperbolic paraboloid
D) , a sphere
E) , a hyperbolic paraboloid
.A)
, an elliptic paraboloidB)
, a sphereC)
, a hyperbolic paraboloidD)
, a sphereE)
, a hyperbolic paraboloid فتح الحزمة
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44
An object is pulled 5 feet horizontally across a floor, using a force of 65 pounds. The direction of the force is above the horizontal. Find the work done. Round your answer to two decimal places.
A) 294.55 ft-lb
B) 322.14 ft-lb
C) 137.35 ft-lb
D) 5.62 ft-lb
E) 336.74 ft-lb
above the horizontal. Find the work done. Round your answer to two decimal places.A) 294.55 ft-lb
B) 322.14 ft-lb
C) 137.35 ft-lb
D) 5.62 ft-lb
E) 336.74 ft-lb
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45
Find a set of symmetric equations of the line through the points and .
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
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46
Convert the point from spherical coordinates to rectangular coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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47
Suppose a plane is flying in the direction . Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places.
A) North of West
B) North of West
C) North of East
D) North of West
E) North of South
. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane? Round your answer to two decimal places. A)
North of WestB)
North of WestC)
North of EastD)
North of WestE)
North of South فتح الحزمة
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48
Find the area of the parallelogram with the vertices , and . Round your answer to two decimal places.
A) 122.00
B) 6.00
C) 11.05
D) 12.00
E) 3.46
, and . Round your answer to two decimal places.A) 122.00
B) 6.00
C) 11.05
D) 12.00
E) 3.46
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49
Find the direction cosines of the vector u given below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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50
Find the volume of the parallelepiped with the following vertices.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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51
Given and find .
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
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52
Suppose a plane flies at a constant groundspeed of 100 miles per hour due east and encounters a 40 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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53
Find the coordinates of the midpoint of the line segment joining the points given below. (-6, 6, 1), (-4, 10, 11)
A) (-5, 2, 6)
B) (1, 2, 6)
C) (1, 8, 6)
D) (1, 8, 5)
E) (-5, 8, 6)
A) (-5, 2, 6)
B) (1, 2, 6)
C) (1, 8, 6)
D) (1, 8, 5)
E) (-5, 8, 6)
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54
Find a set of symmetric equations of the line through the point parallel to the vector .
A)
B)
C)
D)
E)
parallel to the vector .A)
B)
C)
D)
E)
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55
A child applies the brakes on a bicycle by applying a downward force of 20 pounds on the pedal when the crank makes a angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places.
A) 15.32 ft-lb
B) 7.66 ft-lb
C) 91.93 ft-lb
D) 6.43 ft-lb
E) 12.86 ft-lb
angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. Round your answer to two decimal places. A) 15.32 ft-lb
B) 7.66 ft-lb
C) 91.93 ft-lb
D) 6.43 ft-lb
E) 12.86 ft-lb
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56
Convert the following point from cylindrical coordinates to rectangular coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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57
Suppose a plane flies at a constant groundspeed of 300 miles per hour due east and encounters a 50 mile-per-hour wind from the northwest. Both the airspeed and the compass direction must change to for the plane to maintain its groundspeed and eastward direction. Find the airspeed to maintain its groundspeed and eastward direction. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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58
Determine whether the following planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection.
A) The planes are neither parallel nor orthogonal, the angle of intersection is .
B) The planes are parallel.
C) The planes are neither parallel nor orthogonal, the angle of intersection is .
D) The planes are orthogonal.
A) The planes are neither parallel nor orthogonal, the angle of intersection is
.B) The planes are parallel.
C) The planes are neither parallel nor orthogonal, the angle of intersection is
.D) The planes are orthogonal.
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59
Suppose a toy wagon is pulled by exerting a force of 25 pounds on a handle that makes a angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place.
A) 1,200.0 ft-lb
B) 427.5 ft-lb
C) 2,350.2 ft-lb
D) 1,174.6 ft-lb
E) 2,348.2 ft-lb
angle with the horizontal as shown in the figure below. Find the work done in pulling the wagon 50 feet. Round your answer to one decimal place. A) 1,200.0 ft-lb
B) 427.5 ft-lb
C) 2,350.2 ft-lb
D) 1,174.6 ft-lb
E) 2,348.2 ft-lb
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60
Find the cross product of the unit vector .
A)
B)
C)
D)
E)
.A)
B)
C)
D)
E)
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61
Find the direction cosines of the vector u given below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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62
Given the equation of a surface of revolution , find the equation of its generating curve in the xy-plane rotating about the x-axis.
A)
B)
C)
D)
E)
, find the equation of its generating curve in the xy-plane rotating about the x-axis.A)
B)
C)
D)
E)
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63
Find a set of parametric equations of the line through the point parallel to the vector .
A)
B)
C)
D)
E)
parallel to the vector .A)
B)
C)
D)
E)
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64
Identify the following quadric surface.
A) elliptic paraboloid
B) elliptic cone
C) hyperboloid of two sheets
D) hyperboloid of one sheet
E) ellipsoid
A) elliptic paraboloid
B) elliptic cone
C) hyperboloid of two sheets
D) hyperboloid of one sheet
E) ellipsoid
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65
Given , find .
A) -6
B) -24
C) -5
D) 4
E) 26
, find .A) -6
B) -24
C) -5
D) 4
E) 26
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66
Given , find .
A)
B)
C)
D)
E)
, find .A)
B)
C)
D)
E)
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67
Given and find .
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
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68
Given and find .
A)
B)
C)
D)
E)
and find .A)
B)
C)
D)
E)
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69
Find the angle between the vectors for u and v given below.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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70
Find the distance between the point and the line given by the set of parametric equations .
A) /3
B) 1
C) /3
D) 3
E) /1
and the line given by the set of parametric equations . A)
/3 B) 1
C)
/3 D) 3
E)
/1 فتح الحزمة
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71
Given and , find .
A)
B)
C)
D)
E)
and , find .A)
B)
C)
D)
E)
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72
Find a set of parametric equations of the line through the points and .
A)
B)
C)
D)
E)
and .A)
B)
C)
D)
E)
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73
Suppose a 48,000-pound truck is parked on a slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place.
A) 47,070.8 lb
B) 47,436.8 lb
C) 47,313.8 lb
D) 47,270.8 lb
E) 47,280.8 lb
slope as shown in the figure. Assume the only force to overcome is that due to gravity. Find the force perpendicular to the hill. Round your answer to one decimal place. A) 47,070.8 lb
B) 47,436.8 lb
C) 47,313.8 lb
D) 47,270.8 lb
E) 47,280.8 lb
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74
Convert the following point from rectangular coordinates to cylindrical coordinates. Give any angles in radians.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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75
Find an equation in rectangular coordinates for the equation given in cylindrical coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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76
Find an equation in spherical coordinates for the equation given in rectangular coordinates.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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77
Given and , find .
A)
B)
C)
D)
E)
and , find .A)
B)
C)
D)
E)
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78
Both the magnitude and the direction of the force on a crankshaft change as the crankshaft rotates. Find the torque on the crankshaft using the position and data shown in the figure. Round your answer to two decimal places.
A) 2,771.28 ft-lb
B) 277.13 ft-lb
C) 226.27 ft-lb
D) 554.26 ft-lb
E) 160.00 ft-lb
A) 2,771.28 ft-lb
B) 277.13 ft-lb
C) 226.27 ft-lb
D) 554.26 ft-lb
E) 160.00 ft-lb
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79
To carry a 500-pound cylindrical weight, two workers lift on the ends of short ropes tied to an eyelet on the top center of the cylinder. One rope makes a angle away from the vertical and the other makes a angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places.
A)
B)
C)
D)
E)
angle away from the vertical and the other makes a angle as shown in the figure below. Find the vertical component of each worker's force. Round your answer to two decimal places.
A)
B)
C)
D)
E)
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80
Find the angle between the vectors for u and v.
A) 46.88
B) 23.20
C) 66.80
D) 42.59
E) 88.85
A) 46.88
B) 23.20
C) 66.80
D) 42.59
E) 88.85
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