Question: Find an equation for the plane consisting

> A stepped shaft ABC consisting of two solid, circular segments is subjected to torques T1 and T2 acting in opposite directions, as shown in the figure. The larger segment of the shaft has a diameter of d1 = 2.25 in. and a length L1 = 30 in.; the smaller

> A space truss is restrained at joints A, B, and C, as shown in the figure. Load P acts in the 1z direction at joint B and in the 2z directions at joint C. Coordinates of all joints are given in terms of dimension variable L (see figure). Let P = 5 kN and

> A space truss is restrained at joints A, B, and C, as shown in the figure. Load 2P is applied at in the 2x direction at joint A, load 3P acts in the 1z direction at joint B, and load P is applied in the 1z direction at joint C. Coordinates of all joints

> A tubular post of outer diameter d2 is guyed by two cables fitted with turnbuckles (see figure). The cables are tightened by rotating the turnbuckles, producing tension in the cables and compression in the post. Both cables are tightened to a tensile for

> A space truss is restrained at joints O, A, B, and C, as shown in the figure. Load P is applied at joint A and load 2P acts downward at joint C. (a) Find reaction force components Ax, By, and Bz in terms of load variable P. (b) Find the axial force in tr

> A space truss has three-dimensional pin supports at joints O, B, and C. Load P is applied at joint A and acts toward point Q. Coordinates of all joints are given in feet (see figure). (a) Find reaction force components Bx, Bz, and Oz. (b) Find the axial

> Repeat 1.3-10 but use the method of sections to find member forces in AB and DC. Data from Problem 10: Find support reactions at A and B and then use the method of joints to find all member forces. Let b = 3 m and P = 80 kN. y 2P b/2 2

> Repeat 1.3-9 but use the method of sections to find member forces in AC and BD. Data from Problem 9: Find support reactions at A and B and then use the method of joints to find all member forces. Let c = 8 ft and P = 20 kips. -2P -Oc = 80° a O3 =

> Find support reactions at A and B and then use the method of joints to find all member forces. Let b = 3 m and P = 80 kN. y 2P b/2 2P D ec = 80° b/2 Og = 40° = 60° B

> Find support reactions at A and B and then use the method of joints to find all member forces. Let c = 8 ft and P = 20 kips. -2P -Oc = 80° a O3 = 40° = 60° D A B 2P c/2- c/2-

> A plane truss has a pin support at F and a roller support at D (see figure). (a) Find reactions at both supports. (b) Find the axial force in truss member FE. 16 kN 19 kN 3 m 13 kN A 3 m B 3 m D 3 m E 4.5 m f1 m

> A plane truss has a pin support at A and a roller support at E (see figure). (a) Find reactions at all supports. (b) Find the axial force in truss member FE. |3 kips 2 kips 10 ft D 1 kips A 10 ft B 10 ft C 10 ft 15 ft 3 ft G

> Consider the plane truss with a pin support at joint 3 and a vertical roller support at joint 5 (see figure). (a) Find reactions at support joints 3 and 5. (b) Find axial forces in truss members 11 and 13. 20 N 45 N 4 5 6 6. 7 12 11 7 10) 2 m 13 9 6

> Segments AB and BCD of beam ABCD are pin connected at x 5 10 ft. The beam is supported by a pin support at A and roller supports at C and D; the roller at D is rotated by 308 from the x axis (see figure). A trapezoidal distributed load on BC varies in in

> A pressurized circular cylinder has a sealed cover plate fastened with steel bolts (see figure). The pressure p of the gas in the cylinder is 290 psi, the inside diameter D of the cylinder is 10.0 in., and the diameter dB of the b

> A copper alloy pipe with a yield stress sY = 290 MPa is to carry an axial tensile load P = 1500 kN (see figure part a). Use a factor of safety of 1.8 against yielding. (a) If the thickness t of the pipe is one-eighth of its outer diameter, what is the mi

> Use traces to sketch and identify the surface. x = y2 - z2  

> Use traces to sketch and identify the surface. y = z2 - x2

> Use traces to sketch and identify the surface. 3x2 - y2 + 3z2 = 0

> Use traces to sketch and identify the surface. 3x2 + y + 3z2 = 0

> Use traces to sketch and identify the surface. 9y2 + 4z2 = x2 1 36

> Differentiate the function. f (x) = ln(sin2x)

> Use traces to sketch and identify the surface. z2 - 4x2 - y2 = 4

> Use traces to sketch and identify the surface. x2 = 4y2 + z2

> Draw a diagram to show that there are two tangent lines to the parabola that y = x2 pass through the point (0, -4). Find the coordinates of the points where these tangent lines intersect the parabola.

> Use traces to sketch and identify the surface. 4x2 + 9y2 + 9z2 = 36

> Use traces to sketch and identify the surface. x = y2 + 4z2

> (a). Find and identify the traces of the quadric surface 2x2 - y2 + z2 = 1 and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1. (b). If the equation in part (a) is changed to x2 - y2 - z2 = 1, what happens to the gr

> (a). Find and identify the traces of the quadric surface x2 + y2 - z2 = 1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b). If we change the equation in part (a) to x2 - y2 + z2 = 1, how is the graph affected

> Find a vector equation and parametric equations for the line. The line through the point (1, 0, 6) and perpendicular to the plane x + 3y + z = 5

> Find a vector equation and parametric equations for the line. The line through the point (0, 1 4, -10) and parallel to the line x = -1 + 2t, y = 6 - 3t, z = 3 + 9t

> Determine whether each statement is true or false in R3. (a). Two lines parallel to a third line are parallel. (b). Two lines perpendicular to a third line are parallel. (c). Two planes parallel to a third plane are parallel. (d). Two planes perpendicula

> Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t, and x = 1 + 2s, y = 5 + 15s, z = -2 + 6s.

> Differentiate the function. f (x) = sin(ln x)

> Show that the lines with symmetric equations x = y = z and x + 1 = y/2 = z/3 are skew, and find the distance between these lines.

> Where does the normal line to the parabola y = x2 - 1 at the point (-1,0) intersect the parabola a second time? Illustrate with a sketch.

> Find equations of the planes that are parallel to the plane x + 2y - 2z = 1 and two units away from it.

> Show that the distance between the parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is //

> Find the distance between the given parallel planes. 6z = 4y - 2x, 9z = 1 - 3x + 6y

> Which of the following four lines are parallel? Are any of them identical? Li: x= 1 + 6t, y=1 – 3t, z= 12t + 5 L2: x= 1 + 2t, y= t, z =1 + 4t L3: 2x – 2 = 4 – 4y = z + 1 L4: r = (3, 1, 5) + t(4, 2, 8)

> Which of the following four planes are parallel? Are any of them identical? Pi: 3x + 6у — 32 — 6 Р:: 4х — 12у + 82 3D 5 Р:: 9у — 1 + 3х + 62 Р:: z — х + 2у — 2

> Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 - t, z = 2t and intersects this line.

> Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1 - t, z = 2t and intersects this line.

> (a) Find the point at which the given lines intersect: (b). Find an equation of the plane that contains these lines. r = (1, 1, 0) + 1(1, –1, 2) r= (2,0, 2) + s(-1, 1,0)

> Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.

> Differentiate the function. f (x) = x ln x - x

> At what point on the y = 1 + 2ex – 3x curve is the tangent line parallel to the line ?Illustrate by 3x – y = 5 graphing the curve and both lines.

> Find dy/dx by implicit differentiation. x4 + x2y2 + y3 = 5

> Find an equation for the plane consisting of all points that are equidistant from the points (2, 5, 5) and (-6, 3, 1).

> Find symmetric equations for the line of intersection of the planes. z = 2x - y - 5, z = 4x + 3y - 5

> Find symmetric equations for the line of intersection of the planes. 5x - 2y - 2z = 1, 4x + y + z = 6

> (a). Find parametric equations for the line of intersection of the planes and (b). find the angle between the planes. 3x - 2y + z = 1, 2x + y - 3z = 3

> (a). Find parametric equations for the line of intersection of the planes and (b). find the angle between the planes. x + y + z = 1, x + 2y + 2z = 1

> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 2y - z = 2, 2x - 2y + z = 1

> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x + 4y - 3z = 1, 23x + 6y + 7z = 0

> Find direction numbers for the line of intersection of the planes x + y + z = 1 and x + z = 0.

> Find equations of both lines that are tangent to the curve y = x3 – 3x2 + 3x -3 and are parallel to the line 3x – y = 15.

> Where does the line through (-3, 1, 0) and (-1, 5, 6) intersect the plane 2x + y - z = -2?

> Differentiate. y = secθ  tanθ 

> Find the point at which the line intersects the given plane. x = t - 1, y = 1 + 2t, z = 3 - t; 3x - y + 2z = 5

> Find the point at which the line intersects the given plane. x = 2 - 2t, y = 3t, z = 1 + t; x + 2y – z = 7

> Use intercepts to help sketch the plane. 6x + 5y - 3z = 15

> Use intercepts to help sketch the plane. 6x - 3y + 4z = 6

> Use intercepts to help sketch the plane. 3x + y + 2z = 6

> Use intercepts to help sketch the plane. 2x + 5y + z = 10

> Find an equation of the plane. The plane that passes through the line of intersection of the planes x - z = 1 and y + 2z = 3 and is perpendicular to the plane x + y - 2z = 1

> Find an equation of the plane. The plane that passes through the point (1, 5, 1) and is perpendicular to the planes 2x + y - 2z = 2 and x + 3z = 4

> Find an equation of the tangent line to the curve y = x4 + 1 that is parallel to the line 32x – y = 15.

> Find an equation of the plane. The plane that passes through the points (0, -2, 5) and (-1, 3, 1) and is perpendicular to the plane 2z = 5x + 4y

> Find an equation of the plane. The plane that passes through the point (3, 1, 4) and contains the line of intersection of the planes x + 2y + 3z = 1 and 2x - y + z = -3

> Explain why the natural logarithmic function y = ln x is used much more frequently in calculus than the other logarithmic functions y = logb x.

> Find an equation of the plane. The plane that passes through the point (6, -1, 3) and contains the line with symmetric equations x/3 = y + 4 = z/2.

> Find an equation of the plane. The plane that passes through the point (3, 5, -1) and contains the line x = 4 - t, y = 2t - 1, z = -3t.

> Find an equation of the plane. The plane through the points (3, 0, -1), (-2, -2, 3), and (7, 1, -4)

> Find an equation of the plane. The plane through the points (2, 1, 2), (3, -8, 6), and (-2, -3, 1)

> Find an equation of the plane. The plane through the origin and the points (3, -2, 1) and (1, 1, 1)

> Find an equation of the plane. The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)

> Find an equation of the plane. The plane that contains the line x = 1 + t, y = 2 - t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1

> Find an equation of the plane. The plane through the point (3, -2, 8) and parallel to the plane z = x + y

> Find an equation of the plane. The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 - t, z = 3 + 4t

> Find an equation of the plane. The plane through the point (-1, 1 2 , 3) and with normal vector i + 4j + k

> Find an equation of the plane. The plane through the point (5, 3, 5) and with normal vector 2i + j – k.

> Find an equation of the plane. The plane through the origin and perpendicular to the vector 1, −2, 5

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. у — 1 y z - 2 L: 1 -1 3 х — 2 L2: у — 3 y -2 7 || 2.

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. х — 2 L1: у — 3 z - 1 1 -2 -3 х — 3 у+ 4 z - 2 y 3 L2: 1 -7 ||

> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. L1: x = 3 + 2t, y = 4 - t, z = 1 + 3t L2: x = 1 + 4s, y = 3 - 2s, z = 4 + 5s

> Find parametric equations for the line segment from (-2, 18, 31) to (11, -4, 48).

> (a) The curve y = x / (1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.3). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

> Find a vector equation for the line segment from (6, -1, 9) to (7, 6, 0).

> (a). Find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane x - y + 3z = 7 (b). In what points does this line intersect the coordinate planes?

> Find parametric equations and symmetric equations for the line. The line of intersection of the planes x + -y + 3z = 1 and x - y + z = 1

> Find parametric equations and symmetric equations for the line. The line through (2, 1, 0) and perpendicular to both i + j and j + k

> Find parametric equations and symmetric equations for the line. The line through the points (-8, 1, 4) and (3, -2, 4)

> Two tanks are participating in a battle simulation. Tank A is at point (325, 810, 561) and tank B is positioned at point (765, 675, 599). (a). Find parametric equations for the line of sight between the tanks. (b). If we divide the line of sight into 5

> Let L1 be the line through the points (1, 2, 6) and (2, 4, 8). Let L2 be the line of intersection of the planes P1 and P2, where P1 is the plane x - y + 2z + 1 = 0 and P2 is the plane through the points (3, 2, -1), (0, 0, 1), and (1, 2, 1). Calculate the

> Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.

> Let L1 be the line through the origin and the point (2, 0, -1). Let L2 be the line through the points (1, -1, 1) and (4, 1, 3). Find the distance between L1 and L2.

> If v1, v2, and v3 are non-coplanar vectors, let (These vectors occur in the study of crystallography. Vectors of the form n1v1 + n2v2 + n3v3, where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k

> Suppose that a ± 0. (a). If a ∙ b = a ∙ c, does it follow that b = c? (b). If a × b = a × c, does it follow that b = c? (c). If a ∙ b = a ∙ c and a × b = a × c, does it follow that b = c?

> Prove that a.c b. c (а Х Ь) . (с х d) - a·d b.d