- 10.4.1: If you cut a 1-inch square out of each corner of an 8.5-by-11-inch ...
- 10.4.2: The prism at right has equilateral triangle bases with side lengths...
- 10.4.3: A triangular pyramid has a volume of 180 cm3 and a height of 12 cm....
- 10.4.4: A trapezoidal pyramid has a volume of 3168 cm3 ,and its height is 3...
- 10.4.5: The volume of a cylinder is 628 cm3 . Find the radius of the base i...
- 10.4.6: If you roll an 8.5-by-11-inch piece of paper into a cylinder by bri...
- 10.4.7: Sylvia has just discovered that the valve on her cement truck faile...
- 10.4.8: A sealed rectangular container 6 cm by 12 cm by 15 cm is sitting on...
- 10.4.9: To test his assistant, noted adventurer Dakota Davis states that th...
- 10.4.10: Use this information to solve Exercises 1012: Water weighs about 63...
- 10.4.11: Use this information to solve Exercises 1012: Water weighs about 63...
- 10.4.12: Use this information to solve Exercises 1012: Water weighs about 63...
- 10.4.13: A standard juice box holds 8 fluid ounces. A fluid ounce of liquid ...
- 10.4.14: The photo at right shows an ice tray that is designed for a person ...
- 10.4.15: Application An auto tunnel through a mountain is being planned. It ...
- 10.4.16: Developing Proof In the figure at right, ABCE is a parallelogram an...
- 10.4.17: Find the height of this right square pyramid. Give your answer to t...
- 10.4.18: EC is tangent at C. ED is tangent at D. mBC = 76. Find x.
- 10.4.19: Construction Use your geometry tools to construct an inscribed and ...
- 10.4.20: Construction Use your compass and straightedge to construct an isos...
- 10.4.21: M is the midpoint of AC and BD. For each statement, select always (...
Solutions for Chapter 10.4: Volume Problems
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).
Tv = Av + Vo = linear transformation plus shift.
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.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A = CTC = (L.J]))(L.J]))T for positive definite A.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
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.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).