Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Textbook Question
Chapter 61, Problem 2A
Find the area of the shaded portion of this figure. Round your answer to 2 decimal places.
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Consider the vector Field
F(x, y, z) = <3x4, 4xz, - 34z+67
25
Consider also the 3-dimensional region D
bounded by the surface 5 = SUSZ where
S=((x, 4,0): x² +42≤16, the unit disc in the
plane z = 0, with boundary circle C={(x,y,0):
x²+4²=14
•
52=2(x14, 1-x²-4²): x² + y² ≤14, an upside
down paraboloid with the same boundary circlec.
let n denote the outward pointing unit normal
vector on 5. (Note that ǹ is only piecewise conti-
nuous: it is discontinuous along the common.
boundary circle C of S, and s₂; but preceuvise
Continuity is just fine, as it is in Green's theorem).
①Calculate the surface integral (Ends using
a double integral.
S₁
Hint: What are the values of F(x, y, z) and of
plane z =0?
on the
②use previous results to write down the value
of the surface integral (F.nds.
Sz
Let K = R be the field of real numbers. Define the algebra of quaternions H.
Prove that H is isomorphic to the algebra A with basis {1, x, y, z} and multi-
plication given by letting 1 be the unit and
8
X
2x-2
y x+y-z
Z
x+y+z-2
Y
x+y+z-2
2y-2
−x + y + z
Z
x - y + z
x+y+z-2
2z-2
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Chapter 61 Solutions
Mathematics For Machine Technology
Ch. 61 - Plans call for triangle BCDto be sheared off a...Ch. 61 - Find the area of the shaded portion of this...Ch. 61 - Four circles are equally spaced on a bolt circle...Ch. 61 - What is the radius of a circle with a...Ch. 61 - What is the supplement of a 17548'29" angle?Ch. 61 - Prob. 6ACh. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...
Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - Find the unknown area, or diameter for each of the...Ch. 61 - A rectangular steel plate 15.10 inches long and...Ch. 61 - Hydraulic pressure of 705.0 pounds per square inch...Ch. 61 - A circular base is shown. The base is cut from a...Ch. 61 - Find the area of the template shown. Round the...Ch. 61 - A force of 62,125 pounds pulls on a steel rod that...Ch. 61 - A piece shown by the shaded portion is to be cut...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Find the unknown area, radius, or central angle...Ch. 61 - Prob. 33ACh. 61 - Prob. 34ACh. 61 - Prob. 35ACh. 61 - Find the unknown area, radius, or central angle...Ch. 61 - A section of a piece of round stock with a...Ch. 61 - Three pieces, each in the shape of a sector, are...Ch. 61 - Prob. 39ACh. 61 - Prob. 40ACh. 61 - Find the area of each of the segments ACE for...Ch. 61 - Prob. 42ACh. 61 - Prob. 43ACh. 61 - Prob. 44ACh. 61 - Prob. 45ACh. 61 - Prob. 46ACh. 61 - Compute the area of the steel insert (shaded...Ch. 61 - A pattern is shown. a. Find the surface area of...Ch. 61 - The shaded piece shown is cut from a circular...Ch. 61 - Prob. 50A
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- DO NOT WANT AI SOLUTION. Will rate depending on how question is answered. THANK YOU.arrow_forwardDetermine if the columns of the matrix span R4. -7-2 50 -8 5 3 -4 9 6 -21 28 10-2 7 27 6 45 Select the correct choice below and fill in the answer box to complete your choice. A. The columns do not span R4 because none of the columns of A are linear combinations of the other columns of A. B. The columns R4 span because at least of the columns of A is a linear combination of the other columns of A. C. The columns do not span R4 because the reduced echelon form of the augmented matrix is not have a pivot in every row. (Type an exact answer for each matrix element.) D. The columns span R4 because the reduced echelon form of the augmented matrix is every row. (Type an exact answer for each matrix element.) which does which has a pivot inarrow_forward7. (12 pts) This is a pretty problem. Below is given a tangent line to three circles (at points A, B, and C respectively), where each circle is also kissing the other two. If the radius of the smallest circle is r, and the radii of the other two circles are r₁ and r2, derive the following fabulous equation: 1 = 1 + 1 A B Carrow_forward
- In this problem you will use the same vector field from Problem 2, namely F(x, y, z) = (3xy, 4xz, -3yz+6) (where you have already verified that div(F) = 0). Do the following: (a) Calculate a vector potential Ā for F. (b) Check your answer by verifying that curl(A) = F. For (a) you can use the step-by-step method from class. Here is a quick review of that method; you can also consult class notes. Consider a C¹ vector field defined for all (x, y, z) = R³, F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) Any vector field A(x, y, z) = (L(x, y, z), M(x, y, z), N(x, y, z)) which is a solution to the vector differential equation curl(A) = Farrow_forwardConsider the vector field F(x, y, z) = (3xy, 4xz, -3yz+6) Consider also the 3-dimensional region D bounded by the surface S S₁ US2 wherearrow_forwardDon't use ai to answer I will report you answerarrow_forward
- 6. (15 pts) Given is point P in the exterior of a circle. From P, a segment is drawn that is tangent to the circle at a point T, and a secant from P intersects the circle at points A and B. Point K is constructed on PÅ so that PK = PT. Then TK is constructed, intersecting the circle at X. All is shown below. Prove that AX = XB. X B A K P [Hint: angle and arc chase.] Tarrow_forward3. (10 pts) Suppose that AABC is an equilateral triangle and that P is a point in its interior. Perpendiculars are dropped from P to each side of the triangle at points X, Y, and Z. Prove: PX + PY + PZ is always equal to the height of the triangle, no matter where P is, by using area as a tool in your proof! C X Y A Ꮓ B [Hint: you'll need to draw a few extra segments first. Apply the triangle area formula a bunch of times.]arrow_forward4. (12 pts) Given is parallelogram ABCD, and a point P on diagonal AC. Prove that APCB and APCD have the same area.arrow_forward
- No chatgpt plsarrow_forward5. (12 pts) Given is parallelogram ABCD, and point P on diagonal AC. Then, EF is constructed through P parallel to BC, as well as GH through P parallel to DC, as shown. Prove that EBHP and GPFD have the same area. D A E P H G C Barrow_forward1. (17 pts) In general, there's no spectacular relationship between the side lengths of a quadrilateral and the lengths of its diagonals. BUT, if the quadrilateral happens to have all four vertices lying on the same circle, there is something special that happens. And that's the point of this problem: Given quadrilateral ABCD, where A, B, C, and D all lie on the same circle, prove that AB x DC AD × BC = AC × BD To make your life easier, just use the following steps. First, construct AX so that ZDAXZBAP. DX AD (a) Show that CB AC XB AB (b) Show that = DC AC D Χ B P C (c) Use parts (a) and (b) to evaluate DX + XB. Finish the proof!arrow_forward
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