 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 onestory 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
ISBN: 9781559538824
Solutions for Chapter 10.6: Volume of a Sphere
Get Full SolutionsDiscovering Geometry: An Investigative Approach was written by and is associated to the ISBN: 9781559538824. This textbook survival guide was created for the textbook: Discovering Geometry: An Investigative Approach, edition: 4. Since 23 problems in chapter 10.6: Volume of a Sphere have been answered, more than 24948 students have viewed full stepbystep solutions from this chapter. Chapter 10.6: Volume of a Sphere includes 23 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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).

Multiplication Ax
= 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+ (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).

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.

Stiffness matrix
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·

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).