Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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2. A trapezoid is any that has exactly two
sides.
䊳
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
A.II EXERCISES
1. A geometric figure composed of more than one
basic shape is a figure.
3. Volume is a measure of the amount of
occupied by an object.
4. Capacity is a measure of an object’s potential to
or something.
5. Discuss/Explain the methods available to find the
perimeter, area, and volume of composite figures.
6. Discuss/Explain the formula for computing the
volume of a prism. Does the concept apply to the
formula for the volume of a cylinder? Explain.
A-22 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
䊳
DEVELOPING YOUR SKILLS
Compute the area of each trapezoid using the dimensions given.
7. 8.
Use a ruler to estimate the area of each trapezoid using the actual measurements.
9. In square millimeters, 10. In square inches,
Graph paper can be an excellent aid in understanding why perimeter must be measured in linear units, while area is
measured in square units. Suppose the graph paper shown has squares which are 1 cm by 1 cm. Use the grid to
answer each question by counting and by computation.
124 in.
7 ft
25 in.
88 mm
64 mm
3 cm
11. What are the perimeter and area of the rectangle
shown?
13. What are the perimeter and area of the triangle
shown?
Exercises 11 and 13
12. What are the perimeter and area of the trapezoid
shown?
14. What are the perimeter and area of the triangle
shown?
Exercises 12 and 14
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APPENDIX II Geometry Review with Unit Conversions A-23
College Algebra G&M—
15. What is the outer circumference of a circular gear
with a radius (center to teeth) of 2.5 cm? Round to
tenths.
In Exercises 19–26, a composite figure is shown. Construct an appropriate formula and use it to find the perimeter,
area, or volume as indicated.
16. If the inner radius of the gear from Exercise 15
(center to base of teeth) is 2.2 cm, what is the
working depth of the gear (height of gear teeth)?
17. Find the missing length, then compute the area and
perimeter.
36 cm
c
27 cm
18. Find the missing length, then compute the area and
perimeter.
39 in.
89 in.
b
19. Perimeter 20. Perimeter
21. Area 22. Area
23. Area 24. Area
25. Determine the outer perimeter and total area of the
track and field shown.
120 ft
80 ft
9 m
6 m
18 in.
18.4 in.
18.4 in.
6 ft
7 ft
11 ft
12 cm
12 cm
12 cm
7.5 in.7.5 in.
7.5 in.
7.5 in.
26. Find the perimeter and area of
the skirt pattern shown.
27. How many square
inches of paper are
needed to cover the
kite?
28. Find the area of the mainsail
and the area of the jib sail of
the sailboat.
29. Find the length of the missing side.
30. Find the height of the trapezoid.
24 in.
h
36 in.
Area 720 in
2
Perimeter ⫽ 72 cm
24 cm
W
36 in.
9 in.
30 in.
12 in.
12 in.
9 ft
8 ft
Main Jib
4 ft
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For each figure, (a) draw auxiliary lines that partition the figure into basic shapes and label the sides as needed.
(b) Give a verbal model and a formula model that could be used to complete the exercise.
A-24 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
31. Area 32. Volume 33. Volume 34. Perimeter
35. Find the composite volume.
10 in.
10 in.
2 in.
5 in.
7 in.
37. Convert 625 quarts to gallons.
38. Convert 52,600 pounds to tons.
39. Convert 2887.5 in
3
to gallons.
40. How many gallons in a container with a volume of
623.7 in
3
?
41. Convert 6500 m to yards.
42. A mountain has a height of 18,300 ft. What is the
height in kilometers?
43. Convert 300 pounds to kilograms.
44. Convert 95 gallons to liters.
36. Find the composite volume.
6 m
10 m
䊳
WORKING WITH FORMULAS
45. Matching: Place the correct letter in the corresponding blank without using any reference material.
A. perimeter of a square B. perimeter of a rectangle C. perimeter of a triangle
D. area of a triangle E. circumference of a circle F. volume of a cube
G. area of a square H. perimeter of a trapezoid I. volume of a rectangular solid
J. area of a circle K. area of a rectangle L. area of a trapezoid
____ LW ____ ____
____ 4S ____ LWH ____
____ ____ ____ S
2
____ ____ S
3
____
46. Surface Area of a Rectangular Box: The surface area of a
rectangular box is found by summing the areas of all six sides. Find the surface area of a
box 15 in. tall, 8 in. wide, and 3 in. deep.
SA ⴝ 2(LW ⴙ LH ⴙ WH)
h1b
1
⫹ b
2
2
2
2L ⫹ 2W
2rS
1
⫹ S
2
⫹ S
3
⫹ S
4
1
2
BH
r
2
S
1
⫹ S
2
⫹ S
3
8 in.
15 in.
3 in.
Exercise 46
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APPENDIX II Geometry Review with Unit Conversions A-25
College Algebra G&M—
䊳
APPLICATIONS (ROUND TO TENTHS AS NEEDED)
47. Similar triangles: To estimate the height of a
flagpole, Mitchell reasons his shadow must be
proportional to the shadow of the flagpole due to
similar triangles. Mitchell is 4 feet tall and his
shadow measures 7.5 ft. The shadow of the
flagpole measures 60 ft. How high is the pole?
48. Similar triangles: A triangular image measuring
8 in. ⫻ 9 in. ⫻ 10 in. is projected on a screen using
an overhead projector. If the smallest side of the
projected image is 2 feet, what are the other
dimensions of the projected image?
49. Volume of a prism: Using a sheet of canvas that
is 10 ft by 16 ft, a simple tent is made using a
long pole through the middle and pegging the
sides into the ground as shown. If the tent is 4 feet
high at the apex, what is the volume of space
contained in the tent?
50. Volume of a prism: The feeding trough at a pig
farm is a right prism with bases that are isosceles
triangles (two equal sides). The trough is 3 ft wide,
2 ft deep, and 12 ft long. What is its volume? If
each load is 7 ft
3
, how many loads of slop are
needed to fill the trough?
12 ft
2 ft
3 ft
4 ft
16 ft
51. Length of a cable: A radio
tower is secured by cables which
are clamped 21.5 m up the tower
and anchored in the ground 9 m
from its base. If 30 cm lengths
are needed to secure the cable
at each end, how long are the
cables? Round to hundredths
of a meter.
52. Height of a kite: Benjamin
Franklin is flying his kite in a
storm once again and has
let out 200 m of string. John
Adams has walked to a position
directly under the kite and is
150 m from Ben. How high is
the kite to the nearest meter?
53. Unit conversions: A U.S. citizen living in Brazil
wants a Brazilian carpenter to build a wooden chest
3 ft high, 2.5 ft wide, and 4 ft long. If the carpenter
knows only the metric system, what dimensions
should the carpenter be given?
54. Unit conversions: A baseball pitcher at the major-
league level can throw the ball around 145 ft/sec.
What is the speed of the ball in kilometers per hour?
150 m
200 m
?
p
Exercise 47
Exercise 49
Exercise 50
9 m
21.5 m
c
Exercise 51
Exercise 52
Exercise 54
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55. Most economical purchase: Missy’s Famous
Pizza Emporium is running a special—one large
for $8.99 or two mediums for $13.49. If a large
pizza has a 14-in. diameter and a medium pizza has
a 12-in. diameter, which is the better buy (least
expensive per square inch)?
56. Most economical purchase: A large can of
peaches costs 79 and is 15 cm high with a radius
of 6 cm. A small can of peaches costs 39 and is
15 cm high with a radius of 4 cm. Which is the
better buy (least expensive per cubic cm)?
57. Volume of a jacuzzi: A certain Jacuzzi tub is 84 in.
long 60 in. wide and 30 in. deep. How many gallons
of water will it take to fill this tub?
58. Volume of a birdbath: I am trying to fill an outdoor
birdbath, and all I can find is a plastic dish pan
7 in. high, with a 12 in. by 15 in. base. If the birdbath
holds 15 gal, how many trips will have to be
made?
59. Paving a walkway: Current
plans call for building a
circular fountain 6 m in
diameter, surrounded by a
circular walkway 1.5 m wide.
(a) What is the approximate
area of the walkway? (b) If
the concrete for the walkway
is 6 cm deep, what volume of cement must be
used? (c) If the cement costs $125/m
3
with a 7%
sales tax, what is the cost of the materials for this
walkway?
60. Paving a driveway: A driveway and turn-about
has the dimensions shown in the figure. (a) What is
the area of the driveway? (b) If the concrete was
poured to a depth of 4 in., what volume of cement
was used to construct the driveway? (c) If the
Exercise 58
¢
¢
cement cost $3.50 ft
3
and a 5.65% tax was paid,
what was the total cost of the materials used to
complete the driveway?
61. Cost of drywall: After the studs are up, the wall
shown in the figure must be covered in drywall.
(a) How many square feet of drywall are needed?
(b) If drywall is sold only in 4 ft by 8 ft sheets, about
how many sheets are required? (c) If drywall costs
$8.25 per sheet and a 5.75% tax must be paid, what
is the total cost of the material?
62. Cost of baseboards: The dimensions for the living
room/dining room of a home are shown, (a) How
many feet and inches of molding are needed for
the baseboards around the perimeter of the room?
(b) If the molding is only sold in 8-ft lengths,
how many are needed? (c) If the molding costs
$1.74 per foot and sales tax is 5.75%, what is the
total cost of the baseboards?
63. Dimensions of an index card: A popular-sized
index card has a length that is 1 in. less than twice
its width. If the card has an area of 15 in
2
, find the
length and width.
14' 4"
10' 9"
10' 9"
2' 3"
6' 8"
4' 3"
Exercise 62
19 ft
10 ft
15 ft
3 ft
7 ft
Exercise 61
33 ft
32 ft
50 ft
12 ft
Exercise 60
A-26 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
6 m
1.5 m
Exercise 59
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64. Dimensions of a ruler: The plastic ruler that Albert uses for graphing lines has a length that is 1 cm more than
seven times its width. If the ruler has an area of 30 cm
2
, find its length and width.
65. Tracking an oil leak: At an oil storage facility, one of the tanks has a slow leak.
If the tank shown was full to begin with, how many gallons have been lost at the
moment the height of the oil in the tank is 24 ft?
66. Tracking a water leak: A cylindrical water tank has developed a slow leak.
If the tank is standing vertically and is 4 ft tall with a radius of 9 in., how
many gallons have leaked out at the moment the height of the water in the
tank is 3 ft?
APPENDIX II Geometry Review with Unit Conversions A-27
College Algebra G&M—
25 ft
50 ft
Exercise 65
䊳
EXTENDING THE CONCEPT
67. The area (in cm
2
) of the region shown in the figure is
a. b. c.
d. e. f. none of these
68. A 2-quart saucepan is 75% full of leftover soup. Which plastic container will best
store the leftover soup (i.e., with minimum wasted volume)?
a. a hemispheric bowl with radius 3.5 in.
b. a cylinder with radius 3 in. and height 5 in.
c. a rectangular container
d. None of these will hold the soup.
4 in. ⫻ 6 in. ⫻ 3 in.
35 ⫺ xy ⫹ 5x35 ⫹ xy ⫺ 7y
35 ⫹ 7y ⫺ 5x35 ⫹ xy35 ⫺ xy
7 cm
5 cm
x cm
y cm
Exercise 67
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A-28 APPENDIX III More on Synthetic Division
A-28
College Algebra G&M—
Appendix III
More on Synthetic Division
As the name implies, synthetic division simulates the long division process, but in
a condensed and more efficient form. It’s based on a few simple observations of
long division, as noted in the division shown in
Figure AIII.1.
Figure AIII.1 Figure AIII.2
remainder remainder
A careful observation reveals a great deal of repetition, as any term in red is a dupli-
cate of the term above it. In addition, since the dividend and divisor must be written
in decreasing order of degree, the variable part of each term is unnecessary as we can
let the position of each coefficient indicate the degree of the term. In other words,
we’ll agree that
Finally, we know in advance that we’ll be subtracting each partial product, so we can
“distribute the negative,” shown at each stage. Removing the repeated terms and vari-
able factors, then distributing the negative to the remaining terms produces Figure
AIII.2. The entire process can now be condensed by vertically compressing the rows of
the division so that a minimum of space is used (Figure AIII.3).
Figure AIII.3 Figure AIII.4
quotient
dividend dividend
products products
sums remainder
quotient
Further, if we include the lead coefficient in the bottom row (Figure AIII.4), the
coefficients in the top row (in blue) are duplicated and no longer necessary, since the
quotient and remainder now appear in the last row. Finally, note all entries in the prod-
uct row (in red) are five times the sum from the prior column. There is a direct connec-
tion between this multiplication by 5 and the divisor , and in fact, it is the zero of
the divisor that is used in synthetic division ( from ). A simple change
in format makes this method of division easier to use, and highlights the location of the
x ⫺ 5 ⫽ 0x ⫽ 5
x ⫺ 5
1
3
2
⫺7
3
2
⫺7
5
15
105
15
10
x ⫺ 5
冄
1
⫺2
⫺13
⫺17
x ⫺ 5
冄
1
⫺2
⫺13
⫺17
1
3
21
3
2
1
⫺2
⫺13
⫺17
represents the polynomial
1x
3
⫺ 2x
2
⫺ 13x ⫺ 17.
⫺7⫺7
10
⫺ 12x ⫺ 102
2
2x ⫺ 17
15
⫺ 13x
2
⫺ 15x2
33x
2
⫺ 13x
5⫺ 1x
3
⫺ 5x
2
2
x ⫺ 5
冄
1 ⫺2 ⫺13
⫺17
x ⫺ 5
冄
x
3
⫺ 2x
2
⫺ 13x ⫺ 17
1
3
2x
2
⫹
3x ⫹ 2
1x
3
⫺ 2x
2
⫺ 13x ⫺ 172⫼ 1x ⫺ 52
↓
↓
↓
cob19545_app_028-029.qxd 12/21/10 8:03 PM Page 28
divisor and remainder (the blue brackets in Figure AIII.5). Note the process begins by
“dropping the lead coefficient into place” (shown in bold). The full process of syn-
thetic division is shown in Figure AIII.6 for the same exercise.
Figure AIII. 5
divisor (use 5, not ⫺5) →← coefficients of dividend
drop lead coefficient into place →← quotient and remainder
appear in this row
We then multiply this coefficient by the “divisor,” place the result in the next column
and add. In a sense, we “multiply in the diagonal direction,” and “add in the vertical
direction.” Continue the process until the division is complete.
Figure AIII. 6
← coefficients of dividend
multiply by 5
← quotient and remainder
appear in this row
The result is , read from the last row.x
2
⫹ 3x ⫹ 2 ⫹
⫺7
x ⫺ 5
1
3
2
⫺7
5
15
10
1
⫺2
⫺13
⫺175
1
1
⫺2
⫺13
⫺175
APPENDIX III More on Synthetic Division A-29
College Algebra G&M—
↓
↓
↓
cob19545_app_028-029.qxd 12/21/10 8:03 PM Page 29
A-30 APPENDIX IV More on Synthetic Division
A-30
College Algebra G&M—
Appendix IV
More on Matrices
Reduced Row-Echelon Form
A matrix is in reduced row-echelon form if it satisfies the following conditions:
1. All null rows (zeroes for all entries) occur at the bottom of the matrix.
2. The first non-zero entry of any row must be a 1.
3. For any two consecutive, nonzero rows, the leading 1 in the higher row is to the left
of the 1 in the lower row.
4. Every column with a leading 1 has zeroes for all other entries in the column.
Matrices A through D are in reduced row-echelon form.
Where Gaussian elimination places a matrix in row-echelon form (satisfying the first
three conditions), Gauss-Jordan elimination places a matrix in reduced row-echelon form.
To obtain this form, continue applying row operations to the matrix until the fourth condi-
tion above is also satisfied. For a system having a unique solution, the diagonal
entries of the coefficient matrix will be 1’s, with 0’s for all other entries. To illustrate, we’ll
extend Example 4 from Section 7.1 until reduced row-echelon form is obtained.
EXAMPLE 4
䊳
Solving Systems Using the Augmented Matrix and Gauss-Jordan Elimination
Solve using Gauss-Jordan elimination:
matrix form →
The final matrix is in reduced row-echelon form with solution ( , 1, 1) just as in Section 7.1.⫺3
£
100⫺3
0101
0011
§
3R3 ⫹ R1 S R1
⫺4R3 ⫹ R2 S R2
£
10⫺3 ⫺6
0145
0011
§⫺R2 ⫹ R1 S R1£
111⫺1
0145
0011
§
£
111⫺1
0145
0011
§
R3
7
S R3£
111⫺1
0145
0077
§2R2 ⫹ R3 S R3£
111⫺1
0145
0 ⫺2 ⫺1 ⫺3
§
£
111⫺1
0145
0 ⫺2 ⫺1 ⫺3
§⫺1R2£
111⫺1
0 ⫺1 ⫺4 ⫺5
0 ⫺2 ⫺1 ⫺3
§⫺2R1 ⫹ R2 S R2£
111⫺1
21⫺2 ⫺7
0 ⫺2 ⫺1 ⫺3
§
£
111⫺1
21⫺2 ⫺7
0 ⫺2 ⫺1 ⫺3
§R1 4 R2£
21⫺2 ⫺7
111⫺1
0 ⫺2 ⫺1 ⫺3
§•
2x ⫹ y ⫺ 2z ⫽⫺7
x ⫹ y ⫹ z ⫽⫺1
⫺2y ⫺ z ⫽⫺3
•
2x ⫹ y ⫺ 2z ⫽⫺7
x ⫹ y ⫹ z ⫽⫺1
⫺2y ⫺ z ⫽⫺3
3 ⫻ 3
A ⫽ £
010005
000103
000012
§
B ⫽ £
1005
0103
0000
§
C ⫽ £
100 5
013⫺2
000 0
§
D ⫽ £
1050
0120
0001
§
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The Determinant of a General Matrix
To compute the determinant of a general square matrix, we introduce the idea of a
cofactor. For an matrix A, is the cofactor of matrix element
a
ij
, where represents the determinant of the corresponding minor matrix. Note that
is the sum of the row and column of the entry, and if this sum is even,
while if the sum is odd, (this is how the sign table for a
determinant was generated). To compute the determinant of an matrix,
multiply each element in any row or column by its cofactor and add. The result is a
tier-like process in which the determinant of a larger matrix requires computing the
determinant of smaller matrices. In the case of a matrix, each of the minor
matrices will be size , whose determinant then requires the computation of other
determinants. In the following illustration, two of the entries in the first row are
zero for convenience. For
Computing the first determinant gives , the second determinant is 14.
This gives:
⫽ 74
det1A2⫽⫺21⫺162⫹ 31142
3 ⫻ 3⫺163 ⫻ 3
we have: det1A2⫽⫺2
#
1⫺12
1⫹1
†
20⫺2
⫺14 1
⫺32 1
†⫹ 132
#
1⫺12
1⫹3
†
12⫺2
3 ⫺11
0 ⫺31
†
A ⫽ ≥
⫺2030
120⫺2
3 ⫺14 1
0 ⫺32 1
¥,
2 ⫻ 2
3 ⫻ 3
4 ⫻ 4
n ⫻ n3 ⫻ 3
1⫺12
i⫹j
⫽⫺11⫺12
i⫹j
⫽ 1,
i ⫹ j
冟
M
ij
冟
A
ij
⫽ 1⫺12
i⫹j
冟
M
ij
冟
n ⫻ n
APPENDIX IV More on Matrices A-31
College Algebra G&M—
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