Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus
78E79E80E81E82E83E84E85E86E87ESketching a Vector In Exercises 87 und 88, sketch the vector v and write its component form. v lies in the xz-plane, has magnitude 5, and nukes an angle of 45 with die positive z-axis.89E90E92E91E94E95E96E97E98E99E100EAuditorium Lights The lights in an auditorium are 24-pound discs of radius 18 inches. Each disc is supported by three equally spaced cables that are L inches long (see figure). (a) Write the tension T in each cable as a function of L. Determine the domain of the function. (b) Use a graphing utility and the function in part (a) to complete the table. L 20 25 50 35 40 45 50 T (c) Use a graphing utility to graph the function in pan (a). Determine the asymptotes of the graph. (d) Confirm the asymptotes of the graph in part (c) analytically. (e) Determine the minimum length of each cable when a cable is designed to carry a maximum load of 10 pounds.102ELoad Supports Find the tension in each of the supporting cables in the figure when the weight of the crate is 500 new tons.104E105EFinding Dot ProductsIn Exercises 310, find (a) uv (b) uu (c) v2 (d) (uv)v, and (e) u(3v). u=3,4,v=1,52E3E4E5E6E7E8E9EFinding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u= 3,1 ,v= 2,1Finding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u=3i+j,v=2i+4j12EFinding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u= 1,1,1 ,v= 2,1,1Finding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u=3i+2j+k,v=2i3j15E16E17E18EComparing VectorsIn Exercises 2126, determine whether u and v are orthogonal, parallel, or neither. u=4,3 v=12,2320E21EComparing VectorsIn Exercises 2126, determine whether u and v are orthogonal, parallel, or neither. u=2i+3jkv=2ijkComparing VectorsIn Exercises 2126, determine whether u and v are orthogonal, parallel, or neither. u=2,3,1v=1,1,124E25E26E27E28E29E30E31E32E33E34E35EFinding the Projection of u onto v In Exercises 37-44, (a) find the projection of u onto v and (b) find the vector component of u orthogonal to v. u=9,7,v=1,337E38E39E40E41E42E43E44E45E46E47E48EProjection When the projection of u onto v has the same magnitude as the projection of v onto u, can you conclude that u=v? Explain.50ERevenue The vector u= 3240,1450,2235 gives the number of hamburgers, chicken sandwiches and cheeseburgers, respectively, sold at a fast food restaurant in one week. The vector v= 2.25,2.95,2.65 gives the price (in dollars) per unit for the three food items. Find the dot product uv and explain what information it gives.RevenueRepeat Exercises 49 after decreasing the prices by 2%. Identify the vector operation used to decrease the prices by 2%.53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70EProof Use vectors to prove that the diagonals of a rhombus are perpendicular.Proof Use vectors to prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.Bond AngleConsider a regular tetrahedron with vertices (0,0,0),(k,k,0),(k,0,k) and (0,k,k), where k is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid (k/2,k/2,k/2) to two vertices. This is the bond angle for a molecule, such as CH4 (methane) or PbCl4 (lead tetrachloride), where the structure of the molecule is a tetrahedron.74E75EProof Prove the Cauchy-Schwarz Inequality, uv u v .77E78E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25EArea In Exercises 23 and 24, verify that the points are the vertices of a parallelogram, and find its area. A(2,3,1),B(6,5,1),C(7,2,2),D(3,6,4)27E28ETorque The brakes on a bicycle are applied using a downward force of 20 pounds on the pedal when the crank makes a 40 angle with the horizontal (sec figure). The crank is 6 inches in length. Find the torque at P.30EOptimization A force of 180 pounds acts on the bracket shown in the figure. (a) Determine the vector AB and the vector F representing the force. (F will be in terms of .) (b) Find the magnitude of the moment about A by evaluating ABF. (c) Use the result of part (b) to determine the magnitude of the moment when =30. (d) Use the result of part (b) to determine the angle when the magnitude of the moment is maximum. At that angle, what is the relationship between the vectors F and AB? Is it what you expected? Why or why not? (c) Use a graphing utility to graph the function for the magnitude of the moment about A for 0180. Find the zero of the function in the given domain. Interpret the meaning of the zero in the context of the problemOptimization A force of 56 pounds acts on the pipe wrench shown in the figure. (a) Find the magnitude of the moment about O by evaluating OAF. Use a graphing utility to graph the resulting function of . (b) Use the result of part (a) to determine the magnitude of the moment when =45. (c) Use the result of part (a) to determine the angle when the magnitude of the moment is maximum. Is the answer what you expected? Why or why not?33E34E35E36EVolume In Exercises 35 and 36, use t triple scalar product to find the volume of t parallelepiped having adjacent edges u, v, and w. u=i+jv=j+kw=i+kVolume In Exercises 35 and 36, use t triple scalar product to find the volume of t parallelepiped having adjacent edges u, v, and w. u=1,3,1v=0,6,6w=4,0,4Volume In Exercises 37 and 38, find the volume of the parallelepiped with the given vertices. (0,0,0),(3,0,0),(0,5,1),(2,0,5),(3,5,1),(5,0,5),(2,5,6),(5,5,6)Volume In Exercises 37 and 38, find the volume of the parallelepiped with the given vertices. (0,0,0),(0,4,0),(3,0,0),(1,1,5),(3,4,0),(1,5,5),(4,1,5),(4,5,5)41E42E43E44E45E46E47E48E49E50E51E52E53E54EProof In Exercises 47-52, prove the property of the cross product. uv is orthogonal to both u and v.56EProof Prove that uv=uv if u and v are orthogonal.58E59ECONCEPT CHECK Parametric and Symmetric EquationsGive the parametric equations and the symmetric equations of a line in space. Describe what is required to find these equations.1E2E3E4E5EFinding Parametric and Symmetric Equations In Exercises 7-12, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.) Point Parallel to (3,0,2) v=6j+3k7E8E9E10EFinding Parametric and Symmetric Equations In Exercises 13-16, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the two points (if possible). (For each line, write the direction numbers as integers.) (7,2,6),(3,0,6)12E13E14E15E16E17E18E19E20E21E22E23E24EDetermining Parallel Lines In Exercises 29-32, determine whether the lines are parallel or identical. x=63t,y=2+2t,z=5+4tx=6t,y=24t,z=138t26E27EDetermining Parallel Lines In Exercises 29-32, determine whether the lines are parallel or identical. x14=y12=z+24x+21=y10.5=z31Finding a Point of IntersectionIn Exercises 3336, determine whether the lines intersect, and if so, find the point of intersection and the angle between the lines. x=4t+2,y=3,z=t+1x=2s+2,y=2s+3,z=s+130E31E32E33E34E35EFinding an Equation of a PlaneIn Exercises 3944, find an equation of the plane that passes through the given point and is perpendicular to the given vector or line. PointPerpendicular to (0,1,4) n=kFinding an Equation of a PlaneIn Exercises 3944, find an equation of the plane that passes through the given point and is perpendicular to the given vector or line. PointPerpendicular to (3,2,2) n=2i+3jk38E39EFinding an Equation of a PlaneIn Exercises 3944, find an equation of the plane that passes through the given point and is perpendicular to the given vector or line. PointPerpendicular to (3,2,2) x14=y+2=z+33Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (0,0,0),(2,0,3), and (3,1,5).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (3,1,2),(2,1,5), and (1,2,2).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (1,2,3),(3,2,1), and (1,2,2).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the point (1,2,3) and is parallel to the yz-plane.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the point (1,2,3) and is parallel to the xy-plane.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane contains the y-axis and makes an angle of 6 with the positive x-axis.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane contains the lines given by x12=y4=zandx23=y14=z21.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the point (2,2,1) and contains the line given by x2=y41=z.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the points (2,2,1) and (1,1,1) and is perpendicular to the plane 2x3y+z=3.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the points (3,2,1) and (3,1,5) and is perpendicular to the plane 6x+7y+2z=10.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the points (1,2,1) and (2,5,6) and is parallel to the x-axis.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the points (4,2,1) and (3,5,7) and is parallel to the z-axis.Finding an Equation of a PlaneIn Exercises 5760, find an equation of the plane that contains all the points that are equidistant from the given points. (2,2,0),(0,2,2)54E55E56E71E72EParallel PlanesIn Exercises 6164, determine whether the planes are parallel or identical. 3x2y+5z=1075x50y+125z=25074EIntersection of PlanesIn Exercises 6568, (a) find the angle between the two planes and (b) find a set of parametric equations for the line of intersection of the planes. 3x+2yz=7x4y+2z=076E57E58E59E60E61E62E63E64E65E66E67E68E69E70E77E78E79E80E81E82E83E84E85EFinding the Distance Between Two Parallel Planes In Exercises 85-88, verify that the two planes are parallel, and find the distance between the planes. 4x4y+9z=74x4y+9z=1887E88E89E90E91E92E93E94E96E97E98E99EHOW DO YOU SEE IT? Match the general equation with its graph. Then state what axis or plane the equation is parallel to. (a) ax+by+d=0 (b) ax+d=0 (c) cz+d=0 (d) ax+cz+d=0101EMechanical Design The figure shows a chute at the top of a grain elevator of a combine that funnels the grain into a bin. Find the angle between two adjacent sides.DistanceTwo insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point (x,y,z) on the line x=6+t,y=8t,z=3+t. Also, at time t, the second insect is at the point (x,y,z) on the line x=1+t,y=2+t,z=2t. Assume that distances are given in inches. (a) Find the distance between the two insects at time t=0. (b) Use a graphing utility to graph the distance between the insects from t=0 to t=10. (c) Using the graph from part (b), what can you conclude about the distance between the insects? (d) How close to each other do the insects get?104EFinding a Point of IntersectionFind the point of intersection of the plane 3xy+4z=7 and the line through (5,4,3) that is perpendicular to this plane.106EFinding a Point of Intersection Find the point of intersection of the line through (1, -3, 1) and (3, -4, 2) and the plane given by x y + z = 2.108E109ETrue or False? In Exercises 113118, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Two lines in space are either intersecting or parallel.111E112E113E114E1EMatching In Exercises 5-10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (F).] 15x24y2+15z2=43E4E5E6E7E8E9E10E11ESketching a Surface in SpaceIn Exercises 1114, describe and sketch the surface. y2z2=2517E13E14E15E20E16E18E23E19E21E