 3.5.1: Two techniques for fitting models to data are called direct _______...
 3.5.2: Statisticians use a measure called ________ of________ ________ to ...
 3.5.3: The linear model with the least sum of square differences is called...
 3.5.4: An value of a set of data, also called a ________ ________, gives ...
 3.5.5: Direct variation models can be described as varies directly as or i...
 3.5.6: In direct variation models of the form is called the ________ of __...
 3.5.7: The direct variation model can be described as varies directly as t...
 3.5.8: The mathematical model is an example of ________ variation
 3.5.9: Mathematical models that involve both direct and inverse variation ...
 3.5.10: The joint variation model can be described as varies jointly as and...
 3.5.11: EMPLOYMENT The total numbers of people (in thousands) in the U.S. c...
 3.5.12: SPORTS The winning times (in minutes) in the womens 400meter frees...
 3.5.13: 12345 1 2 3 4 5 y
 3.5.14: 12345 1 2 3 4 5 y
 3.5.15: 12345 1 2 3 4 5 y
 3.5.16: 12345 1 2 3 4 5 y
 3.5.17: SPORTS The lengths (in feet) of the winning mens discus throws in t...
 3.5.18: SALES The total sales (in billions of dollars) for CocaCola Enterpr...
 3.5.19: DATA ANALYSIS: BROADWAY SHOWS The table shows the annual gross tick...
 3.5.20: DATA ANALYSIS: TELEVISION SETS The table shows the numbers (in mill...
 3.5.21: 2 4 2 4 y
 3.5.22: 2 4 6 8 2468 y
 3.5.23: k 1
 3.5.24: k 2
 3.5.25: k 1 2
 3.5.26: k 1 4
 3.5.27: k 2
 3.5.28: k 5
 3.5.29: k 10
 3.5.30: k 20
 3.5.31: x 5 10 15 20 25 y 1 1 2 1 3 1 4 1 5
 3.5.32: x 5 10 15 20 25 y 2 4 6 8 10
 3.5.33: x 5 10 15 20 25 y 3.5 7 10.5 14 17.5 x 5
 3.5.34: x 5 10 15 20 25 y 3.5 7 10.5 14 17.5 x 5
 3.5.35: x 5, y 12
 3.5.36: x 2, y 14
 3.5.37: x 10, y 2050
 3.5.38: x 6, y 580
 3.5.39: SIMPLE INTEREST The simple interest on an investment is directly pr...
 3.5.40: SIMPLE INTEREST The simple interest on an investment is directly pr...
 3.5.41: MEASUREMENT On a yardstick with scales in inches and centimeters, y...
 3.5.42: MEASUREMENT When buying gasoline, you notice that 14 gallons of gas...
 3.5.43: TAXES Property tax is based on the assessed value of a property. A ...
 3.5.44: TAXES State sales tax is based on retail price. An item that sells ...
 3.5.45: A force of 265 newtons stretches a spring 0.15 meter (see figure).
 3.5.46: A force of 220 newtons stretches a spring 0.12 meter. What force is...
 3.5.47: The coiled spring of a toy supports the weight of a child. The spri...
 3.5.48: An overhead garage door has two springs, one on each side of the do...
 3.5.49: varies directly as the square of
 3.5.50: varies directly as the cube of
 3.5.51: varies inversely as the square of
 3.5.52: varies inversely as the square root of
 3.5.53: varies directly as and inversely as
 3.5.54: is jointly proportional to the square of and the cube of
 3.5.55: BOYLES LAW: For a constant temperature, the pressure of a gas is in...
 3.5.56: NEWTONS LAW OF COOLING: The rate of change of the temperature of an...
 3.5.57: NEWTONS LAW OF UNIVERSAL GRAVITATION: The gravitational attraction ...
 3.5.58: LOGISTIC GROWTH: The rate of growth of a population is jointly prop...
 3.5.59: Area of a triangle:
 3.5.60: Area of a rectangle:
 3.5.61: Area of an equilateral triangle:
 3.5.62: Surface area of a sphere:
 3.5.63: Volume of a sphere:
 3.5.64: Volume of a right circular cylinder:
 3.5.65: Average speed:
 3.5.66: Free vibrations: kg W r d/t V r 2h V 4 3r3 S 4r 2 A 3s2 4 A lw A 1 ...
 3.5.67: varies directly as when
 3.5.68: varies inversely as when
 3.5.69: is inversely proportional to when
 3.5.70: varies jointly as and when and
 3.5.71: is jointly proportional to and the third power of when and
 3.5.72: varies directly as and inversely as the square of when and
 3.5.73: varies directly as the square of and inversely as when and
 3.5.74: varies jointly as and and inversely as the square of when and ECOLOGY
 3.5.75: A stream with a velocity of mile per hour can move coarse sand part...
 3.5.76: A stream of velocity can move particles of diameter or less. By wha...
 3.5.77: If #28 copper wire (which has a diameter of 0.0126 inch) has a resi...
 3.5.78: A 14foot piece of copper wire produces a resistance of 0.05 ohm. U...
 3.5.79: WORK The work (in joules) done when lifting an object varies jointl...
 3.5.80: MUSIC The frequency of vibrations of a piano string varies directly...
 3.5.81: FLUID FLOW The velocity of a fluid flowing in a conduit is inversel...
 3.5.82: BEAM LOAD The maximum load that can be safely supported by a horizo...
 3.5.83: DATA ANALYSIS: OCEAN TEMPERATURES An oceanographer took readings of...
 3.5.84: DATA ANALYSIS: PHYSICS EXPERIMENT An experiment in a physics lab re...
 3.5.85: DATA ANALYSIS: LIGHT INTENSITY A light probe is located centimeters...
 3.5.86: ILLUMINATION The illumination from a light source varies inversely ...
 3.5.87: In the equation for kinetic energy, the amount of kinetic energy is...
 3.5.88: If the correlation coefficient for a least squares regression line ...
 3.5.89: Discuss how well the data shown in each scatter plot can be approxi...
 3.5.90: WRITING A linear model for predicting prize winnings at a race is b...
 3.5.91: WRITING Suppose the constant of proportionality is positive and var...
 3.5.92: WRITING Suppose the constant of proportionality is positive and var...
 3.5.93: WRITING (a) Given that varies inversely as the square of and is dou...
 3.5.94: CAPSTONE The prices of three sizes of pizza at a pizza shop are as ...
Solutions for Chapter 3.5: Mathematical Modeling and Variation
Full solutions for Algebra and Trigonometry  8th Edition
ISBN: 9781439048474
Solutions for Chapter 3.5: Mathematical Modeling and Variation
Get Full SolutionsChapter 3.5: Mathematical Modeling and Variation includes 94 full stepbystep solutions. Since 94 problems in chapter 3.5: Mathematical Modeling and Variation have been answered, more than 51127 students have viewed full stepbystep solutions from this chapter. Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048474. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra and Trigonometry, edition: 8.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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).

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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