 10.5.1: In 120, find the volume of each figure. If necessary, express answe...
 10.5.2: In 120, find the volume of each figure. If necessary, express answe...
 10.5.3: In 120, find the volume of each figure. If necessary, express answe...
 10.5.4: In 120, find the volume of each figure. If necessary, express answe...
 10.5.5: In 120, find the volume of each figure. If necessary, express answe...
 10.5.6: In 120, find the volume of each figure. If necessary, express answe...
 10.5.7: In 120, find the volume of each figure. If necessary, express answe...
 10.5.8: In 120, find the volume of each figure. If necessary, express answe...
 10.5.9: In 120, find the volume of each figure. If necessary, express answe...
 10.5.10: In 120, find the volume of each figure. If necessary, express answe...
 10.5.11: In 120, find the volume of each figure. If necessary, express answe...
 10.5.12: In 120, find the volume of each figure. If necessary, express answe...
 10.5.13: In 120, find the volume of each figure. If necessary, express answe...
 10.5.14: In 120, find the volume of each figure. If necessary, express answe...
 10.5.15: In 120, find the volume of each figure. If necessary, express answe...
 10.5.16: In 120, find the volume of each figure. If necessary, express answe...
 10.5.17: In 120, find the volume of each figure. If necessary, express answe...
 10.5.18: In 120, find the volume of each figure. If necessary, express answe...
 10.5.19: In 120, find the volume of each figure. If necessary, express answe...
 10.5.20: In 120, find the volume of each figure. If necessary, express answe...
 10.5.21: In 2124, find the surface area of each figure.
 10.5.22: In 2124, find the surface area of each figure.
 10.5.23: In 2124, find the surface area of each figure.
 10.5.24: In 2124, find the surface area of each figure.
 10.5.25: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.26: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.27: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.28: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.29: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.30: In 2530, use two formulas for volume to find the volume of each fig...
 10.5.31: Find the surface area and the volume of the figure shown.
 10.5.32: Find the surface area and the volume of the cement block in the fig...
 10.5.33: Find the surface area of the figure shown.
 10.5.34: A machine produces open boxes using square sheets of metal measurin...
 10.5.35: Find the ratio, reduced to lowest terms, of the volume of a sphere ...
 10.5.36: Find the ratio, reduced to lowest terms, of the volume of a sphere ...
 10.5.37: A cylinder with radius 3 inches and height 4 inches has its radius ...
 10.5.38: A cylinder with radius 2 inches and height 3 inches has its radius ...
 10.5.39: A building contractor is to dig a foundation 12 feet long, 9 feet w...
 10.5.40: What is the cost of concrete for a walkway that is 15 feet long, 8 ...
 10.5.41: A furnace is designed to heat 10,000 cubic feet. Will this furnace ...
 10.5.42: A water reservoir is shaped like a rectangular solid with a base th...
 10.5.43: The Great Pyramid outside Cairo, Egypt, has a square base measuring...
 10.5.44: Although the Eiffel Tower in Paris is not a solid pyramid, its shap...
 10.5.45: You are about to sue your contractor who promised to install a wate...
 10.5.46: Two cylindrical cans of soup sell for the same price. One can has a...
 10.5.47: A circular backyard pool has a diameter of 24 feet and is 4 feet de...
 10.5.48: The tunnel under the English Channel that connects England and Fran...
 10.5.49: Explain the following analogy: In terms of formulas used to compute...
 10.5.50: Explain why a cylinder is not a polyhedron
 10.5.51: Make Sense? In 5154, determine whether each statement makes sense o...
 10.5.52: Make Sense? In 5154, determine whether each statement makes sense o...
 10.5.53: Make Sense? In 5154, determine whether each statement makes sense o...
 10.5.54: Make Sense? In 5154, determine whether each statement makes sense o...
 10.5.55: What happens to the volume of a sphere if its radius is doubled?
 10.5.56: A scale model of a car is constructed so that its length, width, an...
 10.5.57: In 5758, find the volume of the darkly shaded region. If necessary,...
 10.5.58: In 5758, find the volume of the darkly shaded region. If necessary,...
 10.5.59: Find the surface area of the figure shown
Solutions for Chapter 10.5: Volume and Surface Area
Full solutions for Thinking Mathematically  6th Edition
ISBN: 9780321867322
Solutions for Chapter 10.5: Volume and Surface Area
Get Full SolutionsChapter 10.5: Volume and Surface Area includes 59 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 59 problems in chapter 10.5: Volume and Surface Area have been answered, more than 62997 students have viewed full stepbystep solutions from this chapter. Thinking Mathematically was written by and is associated to the ISBN: 9780321867322. This textbook survival guide was created for the textbook: Thinking Mathematically, edition: 6.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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.

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

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 subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.