Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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Evaluate without the aid of a calculator.
73. 74.
75. 76.
77. What perfect square is closest to 78?
78. What perfect cube is closest to ?
Perform the operation indicated without the aid of a
calculator.
79. 80.
81. 82.
83. 84.
85. 86.
87. 88.
89. 90.
91. 92.
93. 94.
3
4
7
8
2
3
16
21
15
1
2
4
5
182
75 115260 12
1121021521122132102
18212
1
4
21
2
3
213
5
8
2
1
1
8
1
3
4
24
5
6
1
1
2
2
0.0762 0.90347.045 9.23
45 154224 1312
71
1
3
641
3
8
B
25
49
B
121
36
Evaluate without a calculator, using the order of
operations.
95.
96.
97. 98.
99. 100.
Evaluate using a calculator (round to hundredths).
101.
102.
103.
104.
12 21122
2
41221322
2122
4 2142
2
41321392
2132
5100
a1
0.078
52
b
52
#
20
2475 a1
0.06
4
b
4
#
10
5162 3
2
9 164
4172 6
2
6 149
a
3
2
b
2
a
9
4
b
B
25
64B
9
16
3
5
#
a
5
3
b
2
15 22
2
16 4
#
2 1
12 10 2 5 132
2
A-12 APPENDIX I The Language, Notation, and Numbers of Mathematics
College Algebra G&M—
䊳
WORKING WITH FORMULAS
105. Pitch diameter:
Mesh gears are used to transfer rotary motion
and power from one shaft to another. The pitch
diameter D of a drive gear is given by the formula
shown, where d is the outer diameter of the gear
and n is the number of teeth on the gear. Find the
pitch diameter of a gear with 12 teeth and an outer
diameter of 5 cm.
D ⴝ
d
#
n
n ⴙ 2
106. Pediatric dosages and Clark’s rule:
The amount of medication prescribed for young
children depends on their weight, height, age, body
surface area and other factors. Clark’s rule is a
formula that helps estimate the correct child’s dose
D
C
based on the adult dose D
A
and the weight W
of the child (an average adult weight of 150 lb is
assumed). Compute a child’s dose if the adult dose
is 50 mg and the child weighs 30 lb.
D
C
ⴝ
D
A
#
W
150
d
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APPENDIX I The Language, Notation, and Numbers of Mathematics A-13
College Algebra G&M—
䊳
EXTENDING THE CONCEPT
113. Here are some historical approximations for
Which one is closest to the true value?
Archimedes: Tsu Ch’ung-chih:
Aryabhata: Brahmagupta: 110
62,832
20,000
355
113
3
1
7
.
114. If and is the product
positive or negative?
115. If and is the quotient
positive or negative?
⫺1A ⫼ B2B 6 0,A 6 0
A
#
1⫺B2B 6 0,A 7 0
cm
31
2
15 cm
10 cm
10 cm
AB
10 cm
5 √3 cm10 √2 cm
cm
22
cm
35
2
䊳
APPLICATIONS
Use positive and negative numbers to model the situation, then compute.
107. Temperature changes:At 6:00
P
.
M
., the temperature
was A cold front moves through that causes the
temperature to drop each hour until midnight.
What is the temperature at midnight?
3°F
50°F.
108. Air conditioning: Most air conditioning systems
are designed to create a drop in the air temperature
each hour. How long would it take to reduce the air
temperature from to 71°?86°
2°
109. Record temperatures: The state of California
holds the record for the greatest temperature swing
between a record high and a record low. The record
high was and the record low was
How many degrees difference are there between the
record high and the record low?
⫺45°F.134°F
110. Cold fronts: In Juneau, Alaska, the temperature was
early one morning. A cold front later moved in
and the temperature dropped by lunchtime.
What was the temperature at lunchtime?
32°F
17°F
Use a calculator and the rational/irrational numbers given to compute.
111. An insect crawls cm along the diagonal of a
square, along the length of a line segment,
and around the circumference of a circle.
What is the total distance traveled (to the nearest
100th of a cm)?
10 cm
31
2
cm
1512
112. Find the distance between points A and B
(rounded to the nearest 10th cm), given that the
square and equilateral triangle have sides of
length 10 cm, the circle has a circumference of
22 cm, and the line segment is cm long.
35
2
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Perimeter Formula Area Formula
Definition and Diagram (linear units or units) (square units or units
2
)
triangle
rectangle
square
trapezoid
circle
a three sided polygon
a quadrilateral with four right angles and
opposite sides parallel
a rectangle with four equal sides
a quadrilateral with one pair of parallel sides
(called bases b
1
and b
2
)
the set of all points lying in a
plane that are an equal distance
(called the radius r) from a
given point (called the center C).
b
1
b
2
h
s
1
s
2
s
4
s
3
s
L
W
h
b
s
1
s
2
s
3
sum of all sides
or
C ⫽ d
C ⫽ 2r
P ⫽ s
1
⫹ s
2
⫹ s
3
⫹ s
4
P ⫽ 4s
P ⫽ 2L ⫹ 2W
P ⫽ s
1
⫹ s
2
⫹ s
3
A ⫽ r
2
A ⫽
h
2
1b
1
⫹ b
2
2
A ⫽ s
2
A ⫽ LW
A ⫽
1
2
bh
LEARNING OBJECTIVES
In Section A.II you will review how to:
A. Find the perimeter and
area of common
geometric figures
B. Compute the volume of
common geometric solids
C. Use unit conversion
factors effectively
College Algebra G&M—
Table A.II.1
r
C
A-14
Appendix II
Geometry Review with Unit Conversions
Developing the ability to use mathematics as a descriptive tool is a major goal of this
text. Without a solid understanding of basic geometry, this goal would be difficult to
achieve—as many of the tasks we perform daily are based on decisions regarding size,
measurement, configuration, and the like.
A. Perimeter and Area Formulas
Basic geometry plays an important role in the application of mathematics. For your
convenience, the most common formulas are collected in Table A.II.1, and a focused
effort should be made to commit them to memory. Note that some of the formulas use
subscripted variables, or a variable with a small case number to the lower right (s
l
, s
2
,
and so on). To help understand the table, we quickly review some fundamental terms
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The formulas and both use the
symbol “ ,” which represents the ratio of a circle’s
circumference to its diameter. We will use a two
decimal approximation in calculations done by
hand: . On many calculators, is the
function to the key and produces a much better
approximation (see Figure A.II.1). When using a
calculator, we most often use all displayed digits and
round only the answer to the desired level of accuracy.
If a problem or application uses a formula, begin by stating the formula rather
than by immediately making any substitutions. This will help to prevent many
careless errors. For Example 1, recall that a trapezoid is a quadrilateral with two
parallel sides.
EXAMPLE 1
䊳
Finding the Area of a Window
A basement window is shaped like an isosceles
trapezoid (base angles equal, opposite sides equal
in length), with a height of 10 in. and bases of
1.5 ft and 2 ft. What is the area of the glass in the
window?
Solution
䊳
Before applying the area formula, all measures must use the same unit. In inches
we have . and .
given formula
substitute 10 in. for
h
, 18 in. for
b
1
and 24 in. for
b
2
simplify
result
The area of the glass in the window is 210 in
2
.
Now try Exercises 7 through 18
䊳
Composite Figures
The largest part of geometric applications, whether in art, construction, or architecture,
involve composite figures, or figures that combine basic shapes. In many cases we are
able to partition or break the figure into more common shapes using an auxiliary line,
or a dashed line drawn to highlight certain features of the diagram. When computing a
perimeter, we use only the exposed, outer edges, much as a soldier would guard the
base camp by marching along the outer edge—the perimeter. For composite figures,
it’s helpful to verbally describe the situation given, creating an English language model
that can easily be translated into an equation model.
⫽ 210 in
2
⫽ 15 in.2142 in.2
⫽
10 in.
2
118 in. ⫹ 24 in.2
A ⫽
h
2
1b
1
⫹ b
2
2
2 ft ⫽ 24 in1.5 ft ⫽ 18 in
2nd
⬇ 3.14
C ⫽ 2rC ⫽ d
APPENDIX II Geometry Review with Unit Conversions A-15
College Algebra G&M—
2 ft
10 in.
1.5 ft
WORTHY OF NOTE
In actual practice, most calculations
are done without using the units of
measure, with the correct units
supplied in the final answer. When
like units occur in an exercise, they
are treated just as the numeric
factors. If they are part of a product,
we write the units with an appropri-
ate exponent as in Example 1. If the
like units occur in the numerator and
denominator, they “cancel” as we
will see in Example 5.
and their meaning. A plane is the infinite extension of length and width along a flat sur-
face. Perimeter is the distance around a two-dimensional figure, or a closed figure that
lies in a plane. Many times these figures are polygons, or closed figures composed of
line segments. The general name for a four-sided polygon is a quadrilateral. A right
angle is an angle measuring 90⬚. A quadrilateral with four right angles is called a
rectangle. Area is a measure of the amount of surface covered by a plane figure, with
the measurement given in square units.
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EXAMPLE 2
䊳
Determining the Perimeter and Area of a Composite Figure
Find the perimeter and area of the composite
Figure A.II.2. Use 3.14.
Solution
䊳
To compute the perimeter, use only the exposed
(outer) edges as shown in Figure A.II.3.
•
verbal model
formula model
simplify
result
The perimeter of the figure is about 23.7 ft.
•
verbal model
formula model
substitute 1.7 for
r
(
d
/2)
simplify
result
The area of the figure is about 30.0 ft
2
.
Now try Exercises 19 through 30
䊳
B. Volume
Volume is a measure of the amount of space occupied by a three-dimensional object
and is measured in cubic units. Some of the more common formulas are given in
Table AII.2.
⬇ 30.0373
⬇ 25.5 ⫹ 4.5373
⬇ 17.5213.42⫹
13.14211.72
2
2
A ⫽ LW ⫹
r
2
2
Total Area ⫽ area of rectangle ⫹ one-half area of circle
⬇ 23.738
⬇ 15 ⫹ 3.4 ⫹ 5.338
⬇ 217.52⫹ 3.4 ⫹
13.1421 3.42
2
P ⫽ 2L ⫹ W ⫹
d
2
Perimeter ⫽ three sides of rectangle ⫹ one-half circle
7.5 ft
3.4 ft
7.5 ft
Perimeter
does not include
these lengths
3.4 ft
Figure A.II.3
⬇
A-16 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
A. You’ve just reviewed
how to find the perimeter and
area of common geometric
figures.
substitute 7.5 for
L
, 3.4 for
W
,
and 3.14 for
7.5 ft
3.4 ft
Figure A.II.2
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the set of all points an
equal distance (called
the radius) from a
given point (called the
center)
union of all line
segments connecting
two congruent circles in
parallel planes, meeting
each at a right angle
union of all line
segments connecting a
given point (vertex) to a
given circle (base) and
whose altitude meets the
center of the base at a
right angle
union of all line
segments connecting a
given point (vertex) to a
given polygon (base)
and whose altitude
meets the center of the
base at a right angle
EXAMPLE 3
䊳
Determining the Volume of a Composite Figure
Sand at a cement
factory is being
dumped from a
conveyor belt into a
pile shaped like a right
circular cone atop a
right circular cylinder
(see figure). How many
cubic feet of sand are
there at the moment the
cone is 6 ft high with a
diameter of 10 ft?
APPENDIX II Geometry Review with Unit Conversions A-17
College Algebra G&M—
Volume Formula
Illustration (cubic units or units
3
)
rectangular
solid
cube
a six-sided solid figure
with opposite faces
congruent and adjacent
faces meeting at right
angles
a rectangular solid with
six congruent, square
faces
Table A.II.2
L
H
W
S
r
r
h
h
r
s
h
sphere
right circular
cylinder
right circular
cone
right square
pyramid
WORTHY OF NOTE
It is again worth noting that
units of measure are treated as
though they were numeric factors.
For the cylinder in Example 3:
. This concept is an
important part of the unit conver-
sions often used in the application
of mathematics.
75 ft
3
#
25 ft
2
#
3 ft ⫽
#
15 ft2
2
#
13 ft2 ⫽
6 ft
10 ft
3 ft
V ⫽ S
3
V ⫽ LWH
V ⫽
1
3
s
2
h
V ⫽
1
3
r
2
h
V ⫽ r
2
h
V ⫽
4
3
r
3
cob19545_app_014-027.qxd 12/22/10 11:59 AM Page 17
substitute 5 for
r
, 3 for
h
1
,
and 6 for
h
2
Solution
䊳
verbal model
formula model
simplify
result (exact form)
There are about 392.7 ft
3
of sand in the pile.
Now try Exercises 31 through 36
䊳
C. Unit Conversion Factors
Unit conversion factors are used to convert from one unit of measure to a related unit,
like ounces to pounds. The procedure used is based on the fact that multiplying any
quantity by the equivalent of 1 does not change its value. Several examples of unit con-
version factors are shown here.
a. b. c.
OR OR OR
A unit factor is set up so that the units you are converting from cancel out, leaving
only the units you are converting to. When doing these conversions, it is helpful to set
up the units first and then write the numeric values with their related unit.
EXAMPLE 4
䊳
Unit Conversions—Feet to Miles
Mt. Everest is 29,035 ft high and is the highest mountain in the world when
measured from see level. Convert 29,035 ft to miles.
Solution
䊳
a. Set up the units of the conversion factor so that “feet” will cancel.
b. Recall that , and write the numeric values with their related
unit: “5280” with feet and “1” with mile. Then simplify the result and state
your answer.
Mount Everest is about 5.5 mi high.
Now try Exercises 37 and 38
䊳
Volume and Capacity
Strictly speaking, there is a distinction between the volume of a three dimensional
object and its capacity. Volume is the amount of space occupied by the object, capacity
is considered to be a measure of the object’s potential for holding or storing something.
⬇ 5.49905303
29,035 ft
1
a
1 mi
5280 ft
b⫽
29,035
5280
mi
5280 ft ⫽ 1 mi
29,035 ft
1
a
mi
ft
b
1 gallon
4 quarts
1 ton
2000 pounds
1 foot
12 inches
4 quarts
1 gallon
2000 pounds
1 ton
12 inches
1 foot
⫽ 125
⫽ 75 ⫹ 50
⫽ 152
2
132⫹
1
3
152
2
162
V ⫽ r
2
h
1
⫹
1
3
r
2
h
2
Total Volume ⫽ volume of cylinder ⫹ volume of cone
A-18 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
B. You’ve just reviewed
how to compute the volume
of common geometric solids.
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Some of the common relationships between units of volume and capacity are given in
the following table.
Volume and Capacity
EXAMPLE 5
䊳
Unit Conversions—Cubic Inches to gallons
While taking inventory of a warehouse, workers find a large
cylindrical paint can with no label. The cylinder measures
13 in. high and has a diameter of 10.5 in. (see figure). How
many gallons of paint does the container hold?
Solution
䊳
a. Compute the volume in cubic inches.
volume of a cylinder
substitute for
r
, 13 for
h
, and 3.14 for
simplify
b. Convert cubic inches to gallons using a unit conversion factor.
set up units
result
There are just under 4.9 gal of paint in the container.
Now try Exercises 39 and 40
䊳
U.S. Units and Metric Units
Since U.S. Customary units and metric units were developed independently of each
other, there is no “direct link” between the two, and we usually look up equivalent val-
ues in a reference book or appendix. Since two equal quantities have a ratio of 1, any
conversion factor needed can be formed as before. The length/distance conversions
shown in Table AII.3 are rounded to three decimal places.
⬇ 4.87 gallons
1125 in
3
⫽
1125 in
3
1
a
1 gal
231 in
3
b
⬇ 1125 in
3
10.5
2
⫽ 5.25 ⬇ 13.14215.252
2
1132
V ⫽ r
2
h
1 mL ⫽ 1 cm
3
1 L ⫽ 1000 cm
3
⫽ 1 dm
3
1 gal ⫽ 231 in
3
APPENDIX II Geometry Review with Unit Conversions A-19
College Algebra G&M—
10.5 in.
13 in.
U.S. to Metric Metric to U.S.
1 inch centimeters 1 centimeter inch
1 foot meter 1 meter feet
1 yard meter 1 meter yards
1 mile kilometers 1 kilometer mile⬇ 0.6214⬇ 1.6093
⬇ 1.0936⫽ 0.9144
⬇ 3.2808⫽ 0.3048
⬇ 0.3937⫽ 2.54
Table A.II.3
Using the left half of the table when doing U.S. to metric conversions and the
right-half when doing metric to U.S. conversions helps to simplify the calculation.
EXAMPLE 6
䊳
Unit Conversions—Kilometers to Miles
The flight distance between New York City and London is 5536 kilometers.
Approximately how many miles is the related flight?
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Solution
䊳
Set up the conversion factor, noting from the table that
conversion factors
set up units
result
The flight distance is approximately 3440 mi.
Now try Exercises 41 through 44
䊳
Conversion factors for weight, volume, and capacity can be found online or in ref-
erence books and applied in the same manner.
Similar Triangles
Another important geometric relationship is that of similar triangles. Two triangles
are similar if corresponding angles are equal. The angles are usually named with capital
letters and the side opposite each angle is named used the related lower case letter
(see Figure A.II.4). Similar triangles have the following useful properties:
The phrase corresponding sides means sides that are in the same relative position in
each triangle. This property enables us to find the length of a missing side by setting up
and solving a proportion. One important application of similar triangles involves
ramps or other inclined planes.
EXAMPLE 7
䊳
Determining the Height of a Ramp
The ramp leading up to a loading dock has
a base of 8 m with vertical braces placed
every 2 m. If the first vertical support is
1.5 m high, how high is the dock?
Solution
䊳
Since the two triangles in question are
similar, we set up and solve a proportion.
Let h represent the height of the dock
(see figure).
corresponding sides are in proportion
substitute known values
clear denominators
solve for
h
The dock is 6 ft high.
Now try Exercises 47 and 48
䊳
6 ⫽ h
12 ⫽ 2h
1.5
2
⫽
h
8
brace height
horizontal length
⫽
dock height
dock length
c
B
a
C
b
A
f
E
d
F
e
D
Similar Triangles
Given 䉭ABC and 䉭DEF as shown.
If ⬔A ⬔D, ⬔B ⬔E and ⬔C ⬔F,
then 䉭ABC and 䉭DEF are similar triangles and
corresponding sides are in proportion.
a
d
b
e
b
e
c
f
c
f
a
d
Figure A.II.4
⬇ 3440 mi
5536 km ⫽
5536 km
1
a
0.6214 miles
1 km
b
1 km ⬇ 0.6214 mi
A-20 APPENDIX II Geometry Review with Unit Conversions
College Algebra G&M—
2 m 2 m
1.5 m
2 m 2 m
h
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Additional Applications
Cubes and rectangular boxes are part of a larger family of solids called prisms. A
prism is any solid figure with bases of the same size and shape (the top and bottom
are both called bases). A right prism has sides which are perpendicular to the
bases, with all cross sections congruent. The volume of a right prism is the area of
its base times its height. This is easily seen in the case of a rectangular box:
.
Volume of a Right Prism
where B represents the area of the base
and h represents the height.
EXAMPLE 8
䊳
Determining the Volume of a Right Prism
The feeding trough at a large cattle ranch is a right prism with trapezoidal bases (see
figure). The trough is 1.4 ft deep and 38 ft long. If the bases of the trapezoid are 2 ft
and 3 ft, how many loads of feed are needed to fill the trough if each load is 54 ft
3
?
Solution
䊳
Notice the trough is a right trapezoidal prism.
a. Find the area of the trapezoidal bases:
area of a trapezoid
substitute 1.4 for
h
, 3 for
b
1
, and 2 for
b
2
result
The area of each base is 3.5 ft
2
.
b. Compute the volume of the trough (the right prism).
substitute 3.5 for
B
, 38 for
h
result
The volume of the trough is 133 ft
3
.
c. Find how many 54-ft
3
loads are needed for 133 ft
3
(divide).
Approximately 2.5 loads of feed are needed to fill the trough.
Now try Exercises 49 through 66
䊳
⬇ 2.463
loads ⬇
133 ft
3
54 ft
3
⫽ 133 ft
3
⫽ 13.521382
volume ⫽ area
#
height V ⫽ Bh
⫽ 3.5 ft
2
⫽
1.4
2
13 ⫹ 22
A ⫽
h
2
1b
1
⫹ b
2
2
38 ft
2 ft
3 ft
1.4 ft
V ⫽ Bh,
V ⫽ LWH ⫽ 1LW2H ⫽ 1area of base2
#
H
APPENDIX II Geometry Review with Unit Conversions A-21
College Algebra G&M—
triangular prism trapezoidal prism
C. You’ve just reviewed
how to use unit conversion
factors effectively.
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