 9.1: Consider the solid shown. a) Does it appear to be a prism? b) Is it...
 9.2: Consider the solid shown. a) Does it appear to be a prism? b) Is it...
 9.3: Consider the hexagonal prism shown in Exercise 1. a) How many verti...
 9.4: Consider the triangular prism shown in Exercise 2. a) How many vert...
 9.5: If each edge of the hexagonal prism in Exercise 1 is measured in ce...
 9.6: If each edge of the triangular prism in Exercise 2 is measured in i...
 9.7: Suppose that each of the bases of the hexagonal prism in Exercise 1...
 9.8: Suppose that each of the bases of the triangular prism in Exercise ...
 9.9: Suppose that each of the bases of the hexagonal prism in Exercise 1...
 9.10: Suppose that each of the bases of the triangular prism in Exercise ...
 9.11: A solid is an octagonal prism. a) How many vertices does it have? b...
 9.12: A solid is a pentagonal prism. a) How many vertices does it have? b...
 9.13: Generalize the results found in Exercises 11 and 12 by answering ea...
 9.14: In the accompanying regular pentagonal prism, suppose that each bas...
 9.15: In the regular pentagonal prism shown above, suppose that each base...
 9.16: For the right triangular prism, suppose that the sides of the trian...
 9.17: For the right triangular prism found in Exercise 16, suppose that t...
 9.18: Given that 100 cm 1 m, find the number of cubic centimeters in 1 cu...
 9.19: Given that 12 in. 1 ft, find the number of cubic inches in 1 cubic ...
 9.20: Find the volume and the surface area of a closed box that has dimen...
 9.21: Find the volume and the surface area of a closed box that has dimen...
 9.22: A cereal box measures 2 in. by 8 in. by 10 in. What is the volume o...
 9.23: The measures of the sides of the square base of a box are twice the...
 9.24: For a given box, the height measures 4 m. If the length of the rect...
 9.25: For the box shown, the total area is 94 cm . Determine the value of x.
 9.26: If the volume of the box is 252 in , find the value of x. (See the ...
 9.27: The box with dimensions indicated is to be constructed of materials...
 9.28: A hollow steel door is 32 in. wide by 80 in. tall by in. thick. How...
 9.29: A storage shed is in the shape of a pentagonal prism. The front rep...
 9.30: A storage shed is in the shape of a trapezoidal prism. Each trapezo...
 9.31: A cube is a right square prism in which all edges have the same len...
 9.32: Use the formulas and drawing in Exercise 31 to find (a) the total a...
 9.33: When the length of each edge of a cube is increased by 1 cm, the vo...
 9.34: The numerical value of the volume of a cube equals the numerical va...
 9.35: The sum of the lengths of all edges of a cube is 60 cm. Find the vo...
 9.36: A concrete pad 4 in. thick is to have a length of 36 ft and a width...
 9.37: Zaidah plans a raised flower bed 2 ft high by 12 ft wide by 15 ft l...
 9.38: In excavating for a new house, a contractor digs a hole in the shap...
 9.39: Kristine creates an open box by cutting congruent squares from the ...
 9.40: As in Exercise 39, find the volume of the box if four congruent squ...
 9.41: Kiannas aquarium is boxshaped with dimensions of 2 ft by 1 ft by 8...
 9.42: The gasoline tank on an automobile is boxshaped with dimensions of...
 9.43: For Exercises 43 to 45, consider the oblique regular pentagonal pri...
 9.44: For Exercises 43 to 45, consider the oblique regular pentagonal pri...
 9.45: For Exercises 43 to 45, consider the oblique regular pentagonal pri...
 9.46: It can be shown that the length of a diagonal of a right rectangula...
 9.47: A diagonal of a cube joins two vertices so that the remaining point...
 9.48: When radii and are placed so that they coincide, a 240 sector of a ...
 9.49: A lawn roller in the shape of a right circular cylinder has a radiu...
 9.50: Derive a formula for the total surface area of the hollowcore sphe...
Solutions for Chapter 9: Surfaces and Solids
Full solutions for Elementary Geometry for College Students  6th Edition
ISBN: 9781285195698
Solutions for Chapter 9: Surfaces and Solids
Get Full SolutionsSince 50 problems in chapter 9: Surfaces and Solids have been answered, more than 4233 students have viewed full stepbystep solutions from this chapter. Elementary Geometry for College Students was written by and is associated to the ISBN: 9781285195698. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Geometry for College Students, edition: 6. Chapter 9: Surfaces and Solids includes 50 full stepbystep solutions.

Column space C (A) =
space of all combinations of the columns of A.

Covariance matrix:E.
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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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).

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.