- 10.6.1: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.2: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.3: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.4: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.5: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.6: In Exercises 16, find the volume of each solid. All measurements ar...
- 10.6.7: What is the volume of the largest hemisphere that you could carve o...
- 10.6.8: A sphere of ice cream is placed onto your ice cream cone. Both have...
- 10.6.9: Application Lickety Split ice cream comes in a cylindrical containe...
- 10.6.10: Find the volume of a spherical shell with an outer diameter of 8 me...
- 10.6.11: Which is greater, the volume of a hemisphere with radius 2 cm or th...
- 10.6.12: A sphere has a volume of 972 in3. Find its radius.
- 10.6.13: A hemisphere has a volume of 18 cm3. Find its radius.
- 10.6.14: The base of a hemisphere has an area of 256 cm2. Find its volume.
- 10.6.15: If the diameter of a students brain is about 6 inches, and you assu...
- 10.6.16: A cylindrical glass 10 cm tall and 8 cm in diameter is filled to 1 ...
- 10.6.17: Application This underground gasoline storage tank is a right cylin...
- 10.6.18: Inspector Lestrade has sent a small piece of metal to the crime lab...
- 10.6.19: City law requires that any one-story commercial building supply a p...
- 10.6.20: Plot A, B, C, and D onto graph paper. A is (3, 5). C is the reflect...
- 10.6.21: Technology Use geometry software to construct a segment AB and its ...
- 10.6.22: Technology Use geometry software to construct a circle. Choose a po...
- 10.6.23: Find w, x, and y.
Solutions for Chapter 10.6: Volume of a Sphere
Full solutions for Discovering Geometry: An Investigative Approach | 4th Edition
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).
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pseudoinverse A+ (Moore-Penrose 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).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).