 1.3.1: If a student saves $35 per week, how long will it take to save enou...
 1.3.2: If a steak sells for $8.15, what is the cost per pound?
 1.3.3: If it takes you 4 minutes to read a page in a book, how many words ...
 1.3.4: By paying $1500 cash and the balance in equal monthly payments, how...
 1.3.5: A salesperson receives a weekly salary of $350. In addition, $15 is...
 1.3.6: You have $250 to spend and you need to purchase four new tires. If ...
 1.3.7: A parking garage charges $2.50 for the first hour and $0.50 for eac...
 1.3.8: An architect is designing a house. The scale on the plan is 1 inch ...
 1.3.9: a. Which is the better value: a 15.3ounce box of cereal for $3.37 ...
 1.3.10: a. Which is the better value: a 12ounce jar of honey for $2.25 or ...
 1.3.11: One person earns $48,000 per year. Another earns $3750 per month. H...
 1.3.12: At the beginning of a year, the odometer on a car read 25,124 miles...
 1.3.13: A television sells for $750. Instead of paying the total amount at ...
 1.3.14: In a basketball game, the Bulldogs scored 34 field goals, each coun...
 1.3.15: Calculators were purchased at $65 per dozen and sold at $20 for thr...
 1.3.16: Pens are bought at $0.95 per dozen and sold in groups of four for $...
 1.3.17: Each day a small business owner sells 200 pizza slices at $1.50 per...
 1.3.18: A college tutoring center pays math tutors $8.15 per hour. Tutors e...
 1.3.19: A car rents for $220 per week plus $0.25 per mile. Find the rental ...
 1.3.20: A college graduate receives a salary of $2750 a month for her first...
 1.3.21: Charlene decided to ride her bike from her home to visit her friend...
 1.3.22: A store received 200 containers of juice to be sold by April 1. Eac...
 1.3.23: A storeowner ordered 25 calculators that cost $30 each. The storeow...
 1.3.24: New York City and Washington, D.C. are about 240 miles apart. A car...
 1.3.25: An automobile purchased for $23,000 is worth $2700 after 7 years. A...
 1.3.26: An automobile purchased for $34,800 is worth $8550 after 7 years. A...
 1.3.27: A vending machine accepts nickels, dimes, and quarters. Exact chang...
 1.3.28: How many ways can you make change for a quarter using only pennies,...
 1.3.29: The members of the Student Activity Council on your campus are meet...
 1.3.30: The members of the Student Activity Council on your campus are meet...
 1.3.31: If you spend $4.79, in how many ways can you receive change from a ...
 1.3.32: If you spend $9.74, in how many ways can you receive change from a ...
 1.3.33: You throw three darts at the board shown. Each dart hits the board ...
 1.3.34: Suppose that you throw four darts at the board shown. With these fo...
 1.3.35: Five housemates (A, B, C, D, and E) agreed to share the expenses of...
 1.3.36: Six houses are spaced equally around a circular road. If it takes 1...
 1.3.37: If a has four true/false questions, in how many ways can there be t...
 1.3.38: There are five people in a room. Each person shakes the hand of eve...
 1.3.39: Five runners, Andy, Beth, Caleb, Darnell, and Ella, are in a onemi...
 1.3.40: Eight teams are competing in a volleyball tournament. Any team that...
 1.3.41: Determine a route whose distance is less than 12 miles for running ...
 1.3.42: Determine a route whose distance exceeds 12 miles for running the e...
 1.3.43: The map shows five western states. Trace a route on the map that cr...
 1.3.44: The layout of a city with land masses and bridges is shown. Trace a...
 1.3.45: Jose, Bob, and Tony are college students living in adjacent dorm ro...
 1.3.46: The figure represents a map of 13 countries. If countries that shar...
 1.3.47: a. Use the properties of a magic square to fill in the missing numb...
 1.3.48: a. Use the properties of a magic square to fill in the missing numb...
 1.3.49: As in sudoku, fill in the missing numbers in the 3by3 square so t...
 1.3.50: The missing numbers in the 4by4 array are onedigit numbers. The ...
 1.3.51: Some numbers in the printing of a division problem have become ille...
 1.3.52: If you know how much was paid for several pounds of steak, find the...
 1.3.53: If you know a persons age, find the year in which that person was b...
 1.3.54: If you know how much you earn each hour, find your yearly income.
 1.3.55: Write your own problem that can be solved using the fourstep proced...
 1.3.56: Polyas four steps in problem solving make it possible for me to sol...
 1.3.57: I used Polyas four steps in problem solving to deal with a personal...
 1.3.58: I find it helpful to begin the problemsolving process by restating...
 1.3.59: When I get bogged down with a problem, theres no limit to the amoun...
 1.3.60: Gym lockers are to be numbered from 1 through 99 using metal number...
 1.3.61: You are on vacation in an isolated town. Everyone in the town was b...
 1.3.62: India Jones is standing on a large rock in the middle of a square p...
 1.3.63: One person tells the truth on Monday, Tuesday, Wednesday, and Thurs...
 1.3.64: (This logic problem dates back to the eighth century.) A farmer nee...
 1.3.65: As in sudoku, fill in the missing numbers along the sides of the tr...
 1.3.66: Solve the sudoku puzzle in the middle of the left column on page 40.
 1.3.67: A version of this problem, called the missing dollar problem, first...
 1.3.68: A firefighter spraying water on a fire stood on the middle rung of ...
 1.3.69: The Republic of Margaritaville is composed of four states: A, B, C,...
 1.3.70: How much will it cost to install bicycle racks on campus to encoura...
 1.3.71: How many new counselors are needed on campus to prevent students fr...
 1.3.72: By how much would taxes in your state have to be increased to cut t...
 1.3.73: Is your local electric company overcharging its customers?
 1.3.74: Should solar heating be required for all new construction in your c...
 1.3.75: Group members should describe a problem in need of a solution. Then...
Solutions for Chapter 1.3: Problem Solving
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 1.3: Problem Solving
Get Full SolutionsThis textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6. Chapter 1.3: Problem Solving includes 75 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Since 75 problems in chapter 1.3: Problem Solving have been answered, more than 70963 students have viewed full stepbystep solutions from this chapter.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

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

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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