 8.7.1: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.2: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.3: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.4: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.5: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.6: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.7: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.8: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.9: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.10: In Exercises 110, find the surface area of each solid. All quadrila...
 8.7.11: Explain how you would find the surface area of this obelisk.
 8.7.12: Application Claudette and Marie are planning to paint the exterior ...
 8.7.13: Construction The shapes of the spinning dishes in the photo are cal...
 8.7.14: Use patty paper, templates, or pattern blocks to create a 33 .42 /3...
 8.7.15: Trace the figure at right. Find the lettered angle measures and arc...
 8.7.16: Application Suppose a circular ranch with a radius of 3 km were div...
 8.7.17: Developing Proof Trace the figure at right. Find the lettered angle...
 8.7.18: If the pattern of blocks continues, what will be the surface area o...
Solutions for Chapter 8.7: Surface Area
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 8.7: Surface Area
Get Full SolutionsSince 18 problems in chapter 8.7: Surface Area have been answered, more than 23319 students have viewed full stepbystep solutions from this chapter. Chapter 8.7: Surface Area includes 18 full stepbystep solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.