 6.8.1: y varies directly as x can be modeled by the equation _____________...
 6.8.2: y varies directly as the nth power of x can be modeled by the equat...
 6.8.3: y varies inversely as x can be modeled by the equation ____________...
 6.8.4: y varies directly as x and inversely as z can be modeled by the equ...
 6.8.5: y varies jointly as x and z can be modeled by the equation ________...
 6.8.6: In the equation S = 8A P , S varies __________________ as A and ___...
 6.8.7: In the equation C = 0.02P1P2 d2 , C varies __________________ as P1...
 6.8.8: C varies jointly as A and T. C = 175 when A = 2100 and T = 4. Find ...
 6.8.9: y varies jointly as a and b, and inversely as the square root of c....
 6.8.10: y varies jointly as m and the square of n, and inversely as p. y = ...
 6.8.11: x varies jointly as y and z.
 6.8.12: x varies jointly as y and the square of z.
 6.8.13: x varies directly as the cube of z and inversely as y.
 6.8.14: x varies directly as the cube root of z and inversely as y.
 6.8.15: x varies jointly as y and z and inversely as the square root of w.
 6.8.16: x varies jointly as y and z and inversely as the square of w.
 6.8.17: x varies jointly as z and the sum of y and w.
 6.8.18: x varies jointly as z and the difference between y and w.
 6.8.19: x varies directly as z and inversely as the difference between y an...
 6.8.20: x varies directly as z and inversely as the sum of y and w.
 6.8.21: An alligators tail length, T, varies directly as its body length, B...
 6.8.22: An objects weight on the moon, M, varies directly as its weight on ...
 6.8.23: The height that a ball bounces varies directly as the height from w...
 6.8.24: The distance that a spring will stretch varies directly as the forc...
 6.8.25: If all men had identical body types, their weight would vary direct...
 6.8.26: On a dry asphalt road, a cars stopping distance varies directly as ...
 6.8.27: The figure shows that a bicyclist tips the cycle when making a turn...
 6.8.28: The water temperature of the Pacific Ocean varies inversely as the ...
 6.8.29: a. A mammals average life span, L, in years, varies inversely as it...
 6.8.30: a. A mammals average life span, L, in years, varies inversely as it...
 6.8.31: a. If a mammal has a life span of 20 years, use the graph to estima...
 6.8.32: a. If a mammal has a life span of 50 years, use the graph to estima...
 6.8.33: Radiation machines, used to treat tumors, produce an intensity of r...
 6.8.34: The illumination provided by a cars headlight varies inversely as t...
 6.8.35: Bodymass index, or BMI, takes both weight and height into account ...
 6.8.36: Ones intelligence quotient, or IQ, varies directly as a persons men...
 6.8.37: The heat loss of a glass window varies jointly as the windows area ...
 6.8.38: Kinetic energy varies jointly as the mass and the square of the vel...
 6.8.39: Sound intensity varies inversely as the square of the distance from...
 6.8.40: Many people claim that as they get older, time seems to pass more q...
 6.8.41: The average number of daily phone calls, C, between two cities vari...
 6.8.42: The force of wind blowing on a window positioned at a right angle t...
 6.8.43: The table shows the values for the current, I, in an electric circu...
 6.8.44: What does it mean if two quantities vary directly?
 6.8.45: In your own words, explain how to solve a variation problem.
 6.8.46: What does it mean if two quantities vary inversely?
 6.8.47: Explain what is meant by combined variation. Give an example with y...
 6.8.48: Explain what is meant by joint variation. Give an example with your...
 6.8.49: z = k1x y2
 6.8.50: z = kx21y
 6.8.51: We have seen that the daily number of phone calls between two citie...
 6.8.52: Use a graphing utility to graph any three of the variation equation...
 6.8.53: Im using an inverse variation equation and I need to determine the ...
 6.8.54: The graph of this direct variation equation has a positive constant...
 6.8.55: When all is said and done, it seems to me that direct variation equ...
 6.8.56: Using the language of variation, I can now state the formula for th...
 6.8.57: In a hurricane, the wind pressure varies directly as the square of ...
 6.8.58: The heat generated by a stove element varies directly as the square...
 6.8.59: Galileos telescope brought about revolutionary changes in astronomy...
 6.8.60: Evaluate: 2 1 2 3 4 2 . (Section 3.5, Example 1)
 6.8.61: Factor completely: x2y  9y  3x2 + 27. (Section 5.6, Example 3)
 6.8.62: Find the degree of 7xy + x2y2  5x3  7. (Section 5.1, Example 1)
 6.8.63: If f(x) = 23x + 12, find f(1).
 6.8.64: If f(x) = 23x + 12, find f(8).
 6.8.65: Use the graph of f(x) = 23x + 12 to identify the functions domain a...
Solutions for Chapter 6.8: Modeling Using Variation
Full solutions for Intermediate Algebra for College Students  6th Edition
ISBN: 9780321758934
Solutions for Chapter 6.8: Modeling Using Variation
Get Full SolutionsChapter 6.8: Modeling Using Variation includes 65 full stepbystep solutions. Since 65 problems in chapter 6.8: Modeling Using Variation have been answered, more than 38398 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Intermediate Algebra for College Students, edition: 6. Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758934.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.