 9.2.1: In 14, use the given figure to find its area in square units.
 9.2.2: In 14, use the given figure to find its area in square units.
 9.2.3: In 14, use the given figure to find its area in square units.
 9.2.4: In 14, use the given figure to find its area in square units.
 9.2.5: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.6: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.7: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.8: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.9: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.10: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.11: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.12: In 512, use Table 9.4 on page 588, along with dimensional analysis,...
 9.2.13: In 1314, use the given figure to find its volume in cubic units.
 9.2.14: In 1314, use the given figure to find its volume in cubic units.
 9.2.15: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.16: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.17: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.18: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.19: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.20: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.21: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.22: In 1522, use Table 9.5 on page 590, along with dimensional analysis...
 9.2.23: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.24: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.25: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.26: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.27: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.28: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.29: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.30: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.31: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.32: In 2332, use Table 9.7 on page 591, along with dimensional analysis...
 9.2.33: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.34: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.35: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.36: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.37: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.38: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.39: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.40: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.41: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.42: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.43: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.44: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.45: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.46: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.47: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.48: In 3348, use Table 9.8 on page 591, along with dimensional analysis...
 9.2.49: The bar graph shows the resident population and the land area of th...
 9.2.50: The bar graph shows the resident population and the land area of th...
 9.2.51: The bar graph shows the resident population and the land area of th...
 9.2.52: The bar graph shows the resident population and the land area of th...
 9.2.53: In 5354, find the population density, to the nearest tenth, for eac...
 9.2.54: In 5354, find the population density, to the nearest tenth, for eac...
 9.2.55: In 5556, use the fact that 1 square mile = 640 acres to find the ar...
 9.2.56: In 5556, use the fact that 1 square mile = 640 acres to find the ar...
 9.2.57: A property that measures 8 hectares is for sale. a. How large is th...
 9.2.58: A property that measures 100 hectares is for sale. a. How large is ...
 9.2.59: In 5962, selecting from square centimeters, square meters, or squar...
 9.2.60: In 5962, selecting from square centimeters, square meters, or squar...
 9.2.61: In 5962, selecting from square centimeters, square meters, or squar...
 9.2.62: In 5962, selecting from square centimeters, square meters, or squar...
 9.2.63: In 6366, select the best estimate for the measure of the area of th...
 9.2.64: In 6366, select the best estimate for the measure of the area of th...
 9.2.65: In 6366, select the best estimate for the measure of the area of th...
 9.2.66: In 6366, select the best estimate for the measure of the area of th...
 9.2.67: A swimming pool has a volume of 45,000 cubic feet. How many gallons...
 9.2.68: A swimming pool has a volume of 66,000 cubic feet. How many gallons...
 9.2.69: A container of grapefruit juice has a volume of 4000 cubic centimet...
 9.2.70: An aquarium has a volume of 17,500 cubic centimeters. How many lite...
 9.2.71: 7172 give the approximate area of some of the worlds largest island...
 9.2.72: 7172 give the approximate area of some of the worlds largest island...
 9.2.73: 7374 give the approximate area of some of the worlds largest island...
 9.2.74: 7374 give the approximate area of some of the worlds largest island...
 9.2.75: 7576 involve dosages of the antiinflammatory drug indomethacin, ad...
 9.2.76: 7576 involve dosages of the antiinflammatory drug indomethacin, ad...
 9.2.77: Describe how area is measured. Explain why linear units cannot be u...
 9.2.78: New Mexico has a population density of 17 people per square mile. D...
 9.2.79: Describe the difference between the following problems: How much fe...
 9.2.80: Describe how volume is measured. Explain why linear or square units...
 9.2.81: For a swimming pool, what is the difference between the following u...
 9.2.82: If there are 10 decimeters in a meter, explain why there are not 10...
 9.2.83: Make Sense? In 8386, determine whether each statement makes sense o...
 9.2.84: Make Sense? In 8386, determine whether each statement makes sense o...
 9.2.85: Make Sense? In 8386, determine whether each statement makes sense o...
 9.2.86: Make Sense? In 8386, determine whether each statement makes sense o...
 9.2.87: Singapore has the highest population density of any country: 46,690...
 9.2.88: Nebraska has a population density of 23.8 people per square mile an...
 9.2.89: A high population density is a condition common to extremely poor a...
 9.2.90: Although Alaska is the least densely populated state, over 90% of i...
 9.2.91: Does an adults body contain approximately 6.5 liters, kiloliters, o...
 9.2.92: Is the volume of a coin approximately 1 cubic centimeter, 1 cubic m...
 9.2.93: If you could select any place in the world, where would you like to...
Solutions for Chapter 9.2: Measuring Area and Volume
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 9.2: Measuring Area and Volume
Get Full SolutionsSince 93 problems in chapter 9.2: Measuring Area and Volume have been answered, more than 63323 students have viewed full stepbystep solutions from this chapter. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. Chapter 9.2: Measuring Area and Volume includes 93 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6.

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!

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.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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!).

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = 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.

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.