 108.1: Vocabulary Describe the endpoints of a radius of a sphere.
 108.2: Find each measurement. Give your answers in terms of .the volume of...
 108.3: Find each measurement. Give your answers in terms of .the volume of...
 108.4: Find each measurement. Give your answers in terms of .the radius of...
 108.5: Food Approximately how many times as great is the volume of the gra...
 108.6: Find each measurement. Give your answers in terms of .the surface a...
 108.7: Find each measurement. Give your answers in terms of .the surface a...
 108.8: Find each measurement. Give your answers in terms of .the volume of...
 108.9: Describe the effect of each change on the given measurement of the ...
 108.10: Describe the effect of each change on the given measurement of the ...
 108.11: Find the surface area and volume of each composite figure. 11
 108.12: Find the surface area and volume of each composite figure. 12
 108.13: Find each measurement. Give your answers in terms of .the volume of...
 108.14: Find each measurement. Give your answers in terms of .the volume of...
 108.15: Find each measurement. Give your answers in terms of .the diameter ...
 108.16: Jewelry The size of a cultured pearl is typicallyindicated by its d...
 108.17: Find each measurement. Give your answers in terms of .the surface a...
 108.18: Find each measurement. Give your answers in terms of .the surface a...
 108.19: Find each measurement. Give your answers in terms of .the volume of...
 108.20: Describe the effect of each change on the given measurement of the ...
 108.21: Describe the effect of each change on the given measurement of the ...
 108.22: Find the surface area and volume of each composite figure.22
 108.23: Find the surface area and volume of each composite figure.23
 108.24: Find the radius of a hemisphere with a volume of 144 cm 3 .
 108.25: Find the circumference of a sphere with a surface area of 60 in 2 .
 108.26: Find the volume of a sphere with a circumference of 36 ft.
 108.27: Find the surface area and volume of a sphere centered at (0, 0, 0) ...
 108.28: Estimation A bead is formed by drilling a cylindrical hole witha 2 ...
 108.29: Sports Find the unknown dimensions of the ball for each sport.29
 108.30: Sports Find the unknown dimensions of the ball for each sport.30
 108.31: Sports Find the unknown dimensions of the ball for each sport.31
 108.32: Sports Find the unknown dimensions of the ball for each sport.32
 108.33: Marine Biology The bathysphere was an early version of a submarine,...
 108.34: Geography Earths radius is approximately 4000 mi. About twothirds ...
 108.35: Astronomy Use the table for Exercises 3538. How many times as great...
 108.36: Astronomy Use the table for Exercises 3538. The sum of the volumes ...
 108.37: Astronomy Use the table for Exercises 3538. Which is greater, the s...
 108.38: Astronomy Use the table for Exercises 3538. How many times as great...
 108.39: Critical Thinking In the figure, the hemisphere and the cylinder bo...
 108.40: Write About It Suppose a sphere and a cube have equal surface areas...
 108.41: This problem will prepare you for the Concept Connection on page 72...
 108.42: A sphere with radius 8 cm is inscribed in a cube. Find the ratio of...
 108.43: What is the surface area of a sphere with volume 10 _23 in 3 ? 8 in...
 108.44: Which expression represents the volume of the composite figureforme...
 108.45: . Food The top of a gumball machine is an 18 in. sphere. The machin...
 108.46: The surface area of a sphere can be used to determine its volume. a...
 108.47: Use the diagram of a sphere inscribed in a cylinder for Exercises 4...
 108.48: Use the diagram of a sphere inscribed in a cylinder for Exercises 4...
 108.49: Write an equation that describes the functional relationship for ea...
 108.50: Write an equation that describes the functional relationship for ea...
 108.51: Find the shaded area. Round to the nearest tenth, if necessary. (Le...
 108.52: Find the shaded area. Round to the nearest tenth, if necessary. (Le...
 108.53: Describe the effect on the volume that results from the given chang...
 108.54: Describe the effect on the volume that results from the given chang...
Solutions for Chapter 108: Spheres
Full solutions for Geometry  1st Edition
ISBN: 9780030923456
Solutions for Chapter 108: Spheres
Get Full SolutionsThis textbook survival guide was created for the textbook: Geometry, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 108: Spheres includes 54 full stepbystep solutions. Since 54 problems in chapter 108: Spheres have been answered, more than 41779 students have viewed full stepbystep solutions from this chapter. Geometry was written by and is associated to the ISBN: 9780030923456.

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).

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!

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.)

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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).

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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·