 7.8.7.1.726: Fill in each blank so that the resulting statement is true. y varie...
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 7.8.7.1.733: Use the fourstep procedure for solving variation problems given on...
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 7.8.7.1.741: Use the fourstep procedure for solving variation problems given on...
 7.8.7.1.742: Use the fourstep procedure for solving variation problems given on...
 7.8.7.1.743: In Exercises 1120, write an equation that expresses each relationsh...
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 7.8.7.1.752: In Exercises 1120, write an equation that expresses each relationsh...
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 7.8.7.1.761: Heart rates and life spans of most mammals can be modeled using inv...
 7.8.7.1.762: Heart rates and life spans of most mammals can be modeled using inv...
 7.8.7.1.763: The figure shows the graph of the inverse variation model that you ...
 7.8.7.1.764: The figure shows the graph of the inverse variation model that you ...
 7.8.7.1.765: Radiation machines, used to treat tumors, produce an intensity of r...
 7.8.7.1.766: The illumination provided by a cars headlight varies inversely as t...
 7.8.7.1.767: Bodymass index, or BMI, takes both weight and height into account ...
 7.8.7.1.768: Ones intelligence quotient, or IQ, varies directly as a persons men...
 7.8.7.1.769: The heat loss of a glass window varies jointly as the windows area ...
 7.8.7.1.770: Kinetic energy varies jointly as the mass and the square of the vel...
 7.8.7.1.771: Sound intensity varies inversely as the square of the distance from...
 7.8.7.1.772: Many people claim that as they get older, time seems to pass more q...
 7.8.7.1.773: The average number of daily phone calls, C, between two cities vari...
 7.8.7.1.774: The force of wind blowing on a window positioned at a right angle t...
 7.8.7.1.775: The table shows the values for the current, I, in an electric circu...
 7.8.7.1.776: What does it mean if two quantities vary directly?
 7.8.7.1.777: In your own words, explain how to solve a variation problem.
 7.8.7.1.778: What does it mean if two quantities vary inversely?
 7.8.7.1.779: Explain what is meant by combined variation. Give an example with y...
 7.8.7.1.780: Explain what is meant by joint variation. Give an example with your...
 7.8.7.1.781: In Exercises 4950, describe in words the variation shown by the giv...
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 7.8.7.1.783: We have seen that the daily number of phone calls between two citie...
 7.8.7.1.784: Use a graphing utility to graph any three of the variation equation...
 7.8.7.1.785: In Exercises 5356, determine whether each statement makes sense or ...
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 7.8.7.1.789: In a hurricane, the wind pressure varies directly as the square of ...
 7.8.7.1.790: The heat generated by a stove element varies directly as the square...
 7.8.7.1.791: Galileos telescope brought about revolutionary changes in astronomy...
 7.8.7.1.792: Solve: 8(2 x) 5x. (Section 2.3, Example 2)
 7.8.7.1.793: Divide: 27x3 8 3x 2 . (Section 5.6, Example 3)
 7.8.7.1.794: Factor: 6x3 6x2 120x. (Section 6.5, Example 2)
 7.8.7.1.795: Exercises 6365 will help you prepare for the material covered in th...
 7.8.7.1.796: Exercises 6365 will help you prepare for the material covered in th...
 7.8.7.1.797: Exercises 6365 will help you prepare for the material covered in th...
Solutions for Chapter 7.8: Modeling Using Variation
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 7.8: Modeling Using Variation
Get Full SolutionsThis textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 7.8: Modeling Using Variation includes 72 full stepbystep solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Since 72 problems in chapter 7.8: Modeling Using Variation have been answered, more than 74794 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.