 10.3.1: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.2: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.3: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.4: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.5: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.6: Find the volume of each solid named in Exercises 16. All measuremen...
 10.3.7: In Exercises 79, express the total volume of each solid with the he...
 10.3.8: In Exercises 79, express the total volume of each solid with the he...
 10.3.9: In Exercises 79, express the total volume of each solid with the he...
 10.3.10: Use the information about the base and height each solid to find th...
 10.3.11: Sketch and label a square pyramid with height H feet and each side ...
 10.3.12: Sketch and label two different circular cones, each with a volume 2...
 10.3.13: Mount Fuji, the active volcano in Honshu, Japan, is 3776 m high and...
 10.3.14: Bretislav has designed a crystal glass sculpture. Part of the piece...
 10.3.15: Jamala has designed a container that she claims will hold 50 in3 . ...
 10.3.16: Find the volume of the solid formed by rotating the shaded figure a...
 10.3.17: Find the volume of the liquid in this right rectangular prism. All ...
 10.3.18: Application A swimming pool is in the shape of this prism. A cubic ...
 10.3.19: Application A landscape architect is building a stone retaining wal...
 10.3.20: As bad as tanker oil spills are, they are only about 12% the 3.5 mi...
 10.3.21: ind the surface area of each of the following polyhedrons. (See the...
 10.3.22: Given the triangle at right, reflect D over AC to D. Then reflect D...
 10.3.23: In each diagram, WXYZ is a parallelogram. Find the coordinates of Y
Solutions for Chapter 10.3: Volume of Pyramids and Cones
Full solutions for Discovering Geometry: An Investigative Approach  4th Edition
ISBN: 9781559538824
Solutions for Chapter 10.3: Volume of Pyramids and Cones
Get Full SolutionsSince 23 problems in chapter 10.3: Volume of Pyramids and Cones have been answered, more than 22104 students have viewed full stepbystep solutions from this chapter. 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. Chapter 10.3: Volume of Pyramids and Cones includes 23 full stepbystep solutions. Discovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

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

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).