 8.4.1: If y varies directly as x and y = 18 when x = 15, find y when x = 20.
 8.4.2: Suppose y varies jointly as x and z. Find y when x = 9 and z = 5, ...
 8.4.3: If y varies inversely as x and y = 14 when x = 12, find x when y =...
 8.4.4: Write a direct variation equation that represents this situation.
 8.4.5: Find the pressure at 60 feet
 8.4.6: It is unsafe for amateur divers to swim where the water pressure is...
 8.4.7: Make a table showing the number of pounds of pressure at various de...
 8.4.8: If y varies directly as x and y = 15 when x = 3, find y when x = 12
 8.4.9: If y varies directly as x and y = 8 when x = 6, find y when x = 15
 8.4.10: Suppose y varies jointly as x and z. Find y when x = 2 and z = 27, ...
 8.4.11: If y varies jointly as x and z and y = 80 when x = 5 and z = 8, fin...
 8.4.12: If y varies inversely as x and y = 5 when x = 10, find y when x = 2.
 8.4.13: If y varies inversely as x and y = 16 when x = 5, find y when x = 20.
 8.4.14: How does the circumference of a circle vary with respect to its rad...
 8.4.15: A map of Alaska is scaled so that 3 inches represents 93 miles. How...
 8.4.16: State whether each equation represents a direct, joint, or inverse ...
 8.4.17: State whether each equation represents a direct, joint, or inverse ...
 8.4.18: State whether each equation represents a direct, joint, or inverse ...
 8.4.19: State whether each equation represents a direct, joint, or inverse ...
 8.4.20: State whether each equation represents a direct, joint, or inverse ...
 8.4.21: State whether each equation represents a direct, joint, or inverse ...
 8.4.22: State whether each equation represents a direct, joint, or inverse ...
 8.4.23: State whether each equation represents a direct, joint, or inverse ...
 8.4.24: State whether each equation represents a direct, joint, or inverse ...
 8.4.25: If y varies directly as x and y = 9 when x is 15, find y when x = 21
 8.4.26: If y varies directly as x and x = 6 when y = 0.5, find y when x = 10.
 8.4.27: Suppose y varies jointly as x and z. Find y when x = _1 2 and z = 6...
 8.4.28: If y varies jointly as x and z and y = _1 8 when x = _1 2 and z = 3...
 8.4.29: If y varies inversely as x and y = 2 when x = 25, find x when y = 40
 8.4.30: If y varies inversely as x and y = 4 when x = 12, find y when x = 5.
 8.4.31: Boyles Law states that when a sample of gas is kept at a constant t...
 8.4.32: Charles Law states that when a sample of gas is kept at a constant ...
 8.4.33: Write an equation to represent the average number of laughs produce...
 8.4.34: Is your equation in Exercise 33 a direct, joint, or inverse variation?
 8.4.35: Assume that members of your household laugh the same number of time...
 8.4.36: Write an equation to represent the amount of meat needed to sustain...
 8.4.37: Is your equation in Exercise 36 a direct, joint, or inverse variation?
 8.4.38: How much meat do three Siberian tigers need for the month of January?
 8.4.39: Paul drove from his house to work at an average speed of 40 miles p...
 8.4.40: Many areas of Northern California depend on the snowpack of the Sie...
 8.4.41: According to Johannes Keplers third law of planetary motion, the ra...
 8.4.42: Write an equation that represents this situation.
 8.4.43: If d is the independent variable and I is the dependent variable, g...
 8.4.44: If two people are viewing the same light source, and one person is ...
 8.4.45: The distance between Earth and the Moon is about 3.84 10 8 meters. ...
 8.4.46: The distance between Earth and the Sun is about 1.5 10 11 meters. T...
 8.4.47: The distance between Earth and the Sun is about 1.5 10 11 meters. T...
 8.4.48: Describe two real life quantities that vary directly with each othe...
 8.4.49: Write a realworld problem that involves a joint variation. Solve t...
 8.4.50: Use the information about variation on page 465 to explain how vari...
 8.4.51: Suppose b varies inversely as the square of a. If a is multiplied b...
 8.4.52: If ab = 1 and a is less than 0, which of the following statements c...
 8.4.53: Determine the equations of any vertical asymptotes and the values o...
 8.4.54: Determine the equations of any vertical asymptotes and the values o...
 8.4.55: Determine the equations of any vertical asymptotes and the values o...
 8.4.56: Simplify each expression.
 8.4.57: Simplify each expression.
 8.4.58: Simplify each expression.
 8.4.59: One estimate for the number of cells in the human body is 100,000,0...
 8.4.60: State the slope and the yintercept of the graph of each equation
 8.4.61: State the slope and the yintercept of the graph of each equation
 8.4.62: State the slope and the yintercept of the graph of each equation
 8.4.63: Identify each function as S for step, C for constant, A for absolut...
 8.4.64: Identify each function as S for step, C for constant, A for absolut...
 8.4.65: Identify each function as S for step, C for constant, A for absolut...
 8.4.66: Identify each function as S for step, C for constant, A for absolut...
 8.4.67: Identify each function as S for step, C for constant, A for absolut...
 8.4.68: Identify each function as S for step, C for constant, A for absolut...
Solutions for Chapter 8.4: Direct, Joint, and Inverse Variation
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 8.4: Direct, Joint, and Inverse Variation
Get Full SolutionsSince 68 problems in chapter 8.4: Direct, Joint, and Inverse Variation have been answered, more than 42027 students have viewed full stepbystep solutions from this chapter. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.4: Direct, Joint, and Inverse Variation includes 68 full stepbystep solutions.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.