 105.1: Vocabulary Describe the endpoints of an axis of a cone.
 105.2: Find the lateral area and surface area of each regular pyramid2
 105.3: Find the lateral area and surface area of each regular pyramid3
 105.4: Find the lateral area and surface area of each regular pyramida reg...
 105.5: Find the lateral area and surface area of each right cone. Give you...
 105.6: Find the lateral area and surface area of each right cone. Give you...
 105.7: Find the lateral area and surface area of each right cone. Give you...
 105.8: Describe the effect of each change on the surface area of the given...
 105.9: Describe the effect of each change on the surface area of the given...
 105.10: Find the surface area of each composite figure.10
 105.11: Find the surface area of each composite figure.11
 105.12: Crafts Anna is making a birthday hat from a patternthat is __34 of ...
 105.13: Find the lateral area and surface area of each regular pyramid.13
 105.14: Find the lateral area and surface area of each regular pyramid.14
 105.15: Find the lateral area and surface area of each regular pyramid.a re...
 105.16: Find the lateral area and surface area of each right cone. Give you...
 105.17: Find the lateral area and surface area of each right cone. Give you...
 105.18: Find the lateral area and surface area of each right cone. Give you...
 105.19: Describe the effect of each change on the surface area of the given...
 105.20: Describe the effect of each change on the surface area of the given...
 105.21: Find the surface area of each composite figure.21
 105.22: Find the surface area of each composite figure.22
 105.23: It is a tradition in England to celebrate May 1st byhanging conesh...
 105.24: Find the surface area of each figure.24
 105.25: Find the surface area of each figure.25
 105.26: Find the surface area of each figure.26
 105.27: Find the surface area of each figure.27
 105.28: This problem will prepare you for the Concept Connection on page 72...
 105.29: Find the radius of a right cone with slant height 21 m and surface ...
 105.30: Find the slant height of a regular square pyramid with base perimet...
 105.31: Find the base perimeter of a regular hexagonal pyramid with slant h...
 105.32: Find the surface area of a right cone with a slant height of 25 uni...
 105.33: Find the surface area of each composite figure.33
 105.34: Find the surface area of each composite figure.34
 105.35: Architecture The Pyramid Arena in Memphis, Tennessee, is a square p...
 105.36: Critical Thinking Explain why the slant height of a regular square ...
 105.37: Write About It Explain why slant height is not defined for an obliq...
 105.38: Which expressions represent the surface areaof the regular square p...
 105.39: A regular square pyramid has a slant height of 18 cm and a lateral ...
 105.40: What is the lateral area of the cone? 360 cm 2 450 cm 2 369 cm 2 16...
 105.41: A frustum of a cone is a part of the cone with two parallel bases.T...
 105.42: A frustum of a pyramid is a part of the pyramid with two parallel b...
 105.43: Use the net to derive the formula for the lateral area ofa right co...
 105.44: State whether the following can be described by a linear function. ...
 105.45: State whether the following can be described by a linear function. ...
 105.46: State whether the following can be described by a linear function. ...
 105.47: A point is chosen randomly in ACEF. Find the probability of each ev...
 105.48: A point is chosen randomly in ACEF. Find the probability of each ev...
 105.49: A point is chosen randomly in ACEF. Find the probability of each ev...
 105.50: Find the surface area of each right prism or right cylinder.Round y...
 105.51: Find the surface area of each right prism or right cylinder.Round y...
 105.52: Find the surface area of each right prism or right cylinder.Round y...
Solutions for Chapter 105: Surface Area of Pyramids and Cones
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 105: Surface Area of Pyramids and Cones
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Geometry, edition: 1. Chapter 105: Surface Area of Pyramids and Cones includes 52 full stepbystep solutions. Geometry was written by and is associated to the ISBN: 9780030923456. Since 52 problems in chapter 105: Surface Area of Pyramids and Cones have been answered, more than 46968 students have viewed full stepbystep solutions from this chapter.

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

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

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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 .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

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).

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.

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.