Math Handbook 833 Additional Problems 858 Solutions for Practice Problems 890 Tables 910 919 ...

113 Pages·2004·14.97 MB·English

832 EM TOC-845813 6/8/04 10:22 PM Page 832
Math Handbook 833
I. Symbols 833 Isolating a Variable
Square Root Property
II. Measurements and
Quadratic Equations
Significant Digits 833–836 Quadratic Formula
Significant Digits Dimensional Calculations
Rounding Dimensional Analysis
Operations with Significant Digits
VII. Graphs of Relations 848–852
III. Fractions,Ratios, The Coordinate Plane
Rates,and Proportions 837–838 Graphing Data to
Fractions Determine a Relationship
Ratios Interpolating and Extrapolating
Rates Interpreting Line Graphs
Proportions Linear Equations
Slope
IV. Exponents,Powers,Roots,
Direct Variation
and Absolute Value 839–841 Inverse Variation
Exponents Quadratic Graphs
Square and Cube Roots
VIII.Geometry and
Operations with Exponents
Absolute Value Trigonometry 853–856
Perimeter,Area,and Volume
V. Scientific Notation 841–843
Area Under a Graph
Large Numbers— Right Triangles
Using Positive Exponents Trigonometric Ratios
Small Numbers— Law of Cosines and Law of Sines
Using Negative Exponents Inverses of Sine,Cosine,and Tangent
Operations with Scientific Notation Graphs of Trigonometric Functions
VI. Equations 843–847 IX. Logarithms 857
Order of Operations Logarithms with Base b
Solving Equations Common Logarithms
Antilogarithms or Inverse Logarithms
Additional Problems 858
Solutions for Practice Problems 890
Tables 910
Color Conventions 910 Heats of Fusion and
Electric Circuit Symbols 910 Vaporization of Some
SI Base Units 911 Common Substances 913
SI Derived Units 911 Coefficients of Thermal
Useful Conversions 911 Expansion at 20°C 914
Physical Constants 912 Speed of Sound
SI Prefixes 912 in Various Media 914
Moments of Inertia for Wavelengths of Visible Light 914
Various Objects 912 Dielectric Constants, K(20°C) 914
Densities of Some The Planets 915
Common Substances 913 The Moon 915
Melting and Boiling Points The Sun 915
of Some Substances 913 Periodic Table of the Elements 916
Specific Heats of Some The Elements 917
Common Substances 913 Safety Symbols 918
919
927
832
833-857 EM MHB-845813 6/8/04 10:24 PM Page 833
I. Symbols
(cid:1) change in quantity a(cid:7)b }
(cid:2) plus or minus a quantity ab amultiplied by b
∝ is proportional to a(b)
(cid:3) is equal to a(cid:8)b }
(cid:1) is approximately equal to a/b adivided by b
(cid:2) is approximately equal to (cid:9)a(cid:9)
b
(cid:4) is less than or equal to (cid:3)a(cid:4) square root of a
(cid:5) is greater than or equal to a absolute value of a
(cid:6)(cid:6) is much less than log x log to the base, b,of x
b
≡ is defined as
II. Measurements and Significant Digits
Connecting Math to Physics Math is the language of physics. Using math, physicists
are able to describe relationships among the measurements that they make using equa-
tions. Each measurement is associated with a symbol that is used in physics equations.
The symbols are called variables.
Significant Digits
All measured quantities are approximated and have significant digits. The number
of significant digits indicates the precision of the measurement. Precision is a measure
of exactness. The number of significant digits in a measurement depends on the smallest
unit on the measuring tool. The digit farthest to the right in a measurement is estimated.
Example: What is the estimated digit for each of the measuring sticks in the figure below
used to measure the length of the rod?
Using the lower measuring tool, the length is between 9 and 10 cm. The measurement
would be estimated to the nearest tenth of a centimeter. If the length was exactly on the
9-cm or 10-cm mark, record it as 9.0 cm or 10.0 cm.
Using the upper measuring tool, the length is between 9.5 and 9.6 cm. The measurement
would be estimated to the nearest hundredth of a centimeter. If the length was exactly
on the 9.5-cm or 9.6-cm mark, record it as 9.50 cm or 9.60 cm.
0 mm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
833
833-857 EM MHB-845813 3/22/04 8:43 AM Page 834
All nonzero digits in a measurement are significant digits. Some zeros are significant
k and some are not. All digits between and including the first nonzero digit from the left
o
through the last digit on the right are significant. Use the following rules when determining
o
b the number of significant digits.
d
n
a 1. Nonzero digits are significant.
H
2. Final zeros after a decimal point are significant.
h
t
a 3. Zeros between two significant digits are significant.
M
4. Zeros used only as placeholders are not significant.
Example:State the number of significant digits in each measurement.
5.0 g has two significant digits. Using rules 1 and 2
14.90 g has four significant digits. Using rules 1 and 2
0.0 has one significant digit. Using rules 2 and 4
300.00 mm has five significant digits. Using rules 1, 2, and 3
5.06 s has three significant digits. Using rules 1 and 3
304 s has three significant digits. Using rules 1 and 3
0.0060 mm has two significant digits (6 and the last 0). Using rules 1, 2, and 4
140 mm has two significant digits (just 1 and 4). Using rules 1 and 4
1. State the number of significant digits in each measurement.
a. 1405 m d. 12.007 kg
b. 2.50 km e. 5.8(cid:1)106 kg
c. 0.0034 m f. 3.03(cid:1)10(cid:2)5 mL
There are two cases in which numbers are considered exact, and thus, have an
infinite number of significant digits.
1. Counting numbers have an infinite number of significant digits.
2. Conversion factors have an infinite number of significant digits.
Examples:
The factor “2” in 2mg has an infinite number.
The number 2 is a counting number. It is an exact integer.
The number “4” in 4 electrons has an infinite number.
Because you cannot have a partial electron, the number 4, a counting number,
is considered to have an infinite number of significant digits.
60 s/1 min has an infinite number.
There are exactly 60 seconds in 1 minute, thus there are an infinite number
of significant digits in the conversion factor.
834 Appendix A Math Handbook
833-857 EM MHB-845813 6/8/04 10:26 PM Page 835
Rounding
A number can be rounded to a specific place value (like hundreds or tenths) or to M
a specific number of significant digits. To do this, determine the place being rounded, a
t
h
and then use the following rules.

H
1. When the leftmost digit to be dropped is less than 5, that digit and any digits that a
n
follow are dropped. Then the last digit in the rounded number remains unchanged. d
b
2. When the leftmost digit to be dropped is greater than 5, that digit and any digits that o
o
follow are dropped, and the last digit in the rounded number is increased by one.
k
3. When the leftmost digit to be dropped is 5 followed by a nonzero number, that digit
and any digits that follow are dropped. The last digit in the rounded number increas-
es by one.
4. If the digit to the right of the last significant digit is equal to 5, and 5 is followed by a
zero or no other digits, look at the last significant digit. If it is odd, increase it by one;
if it is even, do not round up.
Examples: Round the following numbers to the stated number of significant digits.
8.7645 rounded to 3 significant digits is 8.76. Using rule 1
8.7676 rounded to 3 significant digits is 8.77. Using rule 2
8.7519 rounded to 2 significant digits is 8.8. Using rule 3
92.350 rounded to 3 significant digits is 92.4. Using rule 4
92.25 rounded to 3 significant digits is 92.2. Using rule 4
2. Round each number to the number of significant digits shown in
parentheses.
a. 1405 m (2) c. 0.0034 m (1)
b. 2.50 km (2) d. 12.007 kg (3)
Operations with Significant Digits
When using a calculator, do all of the operations with as much precision as the
calculator allows, and then round the result to the correct number of significant digits.
The correct number of significant digits in the result depends on the measurements
and on the operation.
Addition and subtraction Look at the digits to the right of the decimal point. Round the
result to the least precise value among the measurements—the smallest number of digits
to the right of the decimal points.
Example: Add 1.456 m, 4.1 m, and 20.3 m.
The least precise values are 4.1 m and 20.3 m because they have only one digit to the
right of the decimal points.
1.456 m
4.1 m
(cid:1)20.3 m Add the numbers
(cid:2)(cid:2)(cid:2)(cid:2)
25.856 m
The sum is only as precise as the least precise number being added.
25.9 m Round the result to the estimated place with the largest place value
Math Handbook 835
833-857 EM MHB-845813 3/22/04 8:44 AM Page 836
Multiplication and division Look at the number of significant digits in each measure-
k ment. Perform the calculation. Round the result so that it has the same number of
o
significant digits as the measurement with the least number of significant digits.
o
b
d Example: Multiply 20.1 m by 3.6 m.
n
a
H (20.1 m)(3.6 m)(cid:1)72.36 m2

h The least precise value is 3.6 m with two significant digits. The product can only have
t
a
M as many digits as the least precise of the multiplied numbers.
72 m Round the result to two significant digits
3. Simplify the following expressions using the correct number
of significant digits.
a. 5.012 km(cid:2)3.4 km(cid:2)2.33 km
b. 45 g(cid:3)8.3 g
c. 3.40 cm (cid:4)7.125 cm
d. 54 m(cid:5)6.5 s
Combination When doing a calculation that requires a combination of addition/
subtraction and multiplication/division, use the multiplication/division rule.
Examples:
d(cid:1)19 m(cid:2)(25.0 m/s)(2.50 s)(cid:2)(cid:6)1(cid:6)((cid:3)10.0 m/s2)(2.50 s)2
2
(cid:1)5.0(cid:4)101 m 19 m only has two significant digits, so the answer should only
have two significant digits
slope(cid:1) (cid:6)70.0 m(cid:3)(cid:6)10.0m
29s(cid:3)11s
(cid:1) 3.3 m/s 26 s and 11 s only have two significant digits each,
so the answer should only have two significant digits
Multistep calculations Do not round to significant digits in the middle of a multistep
calculation. Instead, round to a reasonable number of decimal places that will not cause
you to lose significance in your answer. When you get to your final step where you are
solving for the answer asked for in the question, you should then round to the correct
number of significant digits.
Example:
F(cid:1)(cid:1)((cid:2)24 N)2(cid:2)(cid:2)(36(cid:2) N)2
(cid:1)(cid:1)5(cid:2)76 N2(cid:2)(cid:2)1296(cid:2) N2 Do not round to 580 N2and 1300 N2
(cid:1)(cid:1)1(cid:2)872 N2(cid:2) Do not round to 1800 N2
(cid:1)43 N Final answer, so it should be rounded to two significant digits
836 Appendix A Math Handbook
833-857 EM MHB-845813 6/8/04 10:28 PM Page 837
III. Fractions, Ratios, Rates, and Proportions
M
Fractions
a
A fraction names a part of a whole or a part of a group. A fraction also can express th

a ratio (see page 838). It consists of a numerator, a division bar, and a denominator. H
a
(cid:1)nume(cid:1)rator (cid:5) (cid:1)number (cid:1)of parts(cid:1) chosen nd
denominator total number of parts b
o
o
Simplification Sometimes, it is easier to simplify an expression before substituting k
the known values of the variables. Variables often cancel out of the expression.
pn
Example: Simplify (cid:1)(cid:1).
pw
(cid:1)p(cid:1)n (cid:5)((cid:1)p(cid:1))((cid:1)n(cid:1)) Factor out the pin the numerator and the denominator, and break
pw p w the fraction into the product of two fractions
(cid:5)(1)((cid:1)n(cid:1)) Substitute (p/p)(cid:1)1
w
(cid:5)(cid:1)n(cid:1)
w
Multiplication and division To multiply fractions, multiply the numerators and multiply
the denominators.
s t
Example: Multiply the fractions (cid:1)(cid:1)and (cid:1)(cid:1).
a b
((cid:1)s(cid:1))((cid:1)t(cid:1))(cid:5)(cid:1)s(cid:1)t
Multiply the numerators and the denominators
a b ab
To divide fractions, multiply the first fraction by the reciprocal of the second fraction.
To find the reciprocal of a fraction, invert it—switch the numerator and the denominator.
s t
Example: Divide the fraction (cid:1)(cid:1) by (cid:1)(cid:1).
a b
(cid:1)s(cid:1)(cid:4)(cid:1)t(cid:1)(cid:5)((cid:1)s(cid:1))((cid:1)b(cid:1))
Multiply the first fraction by the reciprocal of the second fraction
a b a t
(cid:5)(cid:1)sb(cid:1)
Multiply the numerators and the denominators
at
Addition and subtraction To add or subtract two fractions, first write them as
fractions with a common denominator; that is, with the same denominator. To find a
common denominator, multiply the denominators of both fractions. Then, add or subtract
the numerators and use the common denominator.
1 2
Example: Add the fractions (cid:1)(cid:1) and (cid:1)(cid:1).
a b
(cid:1)1(cid:1)(cid:2)(cid:1)2(cid:1)(cid:5)((cid:1)1(cid:1))((cid:1)b(cid:1))(cid:2)((cid:1)2(cid:1))((cid:1)a(cid:1))
Multiply each fraction by a fraction equal to 1
a b a b b a
(cid:5)(cid:1)b(cid:1)(cid:2)(cid:1)2(cid:1)a
Multiply the numerators and the denominators
ab ab
(cid:5)(cid:1)b(cid:2)(cid:1)2a
Write a single fraction with the common denominator
ab
4. Perform the indicated operation. Write the answer in simplest form.
a. (cid:1)1(cid:1)(cid:2)(cid:1)y(cid:1) c. ((cid:1)3(cid:1))((cid:1)1(cid:1))
x 3 x y
b. (cid:1)a(cid:1)(cid:3)(cid:1)3(cid:1) d. (cid:1)2(cid:1)a (cid:4)(cid:1)1(cid:1)
2b b 5 2
Math Handbook 837
833-857 EM MHB-845813 6/8/04 8:06 PM Page 838
Ratios
k A ratio is a comparison between two numbers by division. Ratios can be written
o
o in several different ways. The ratio of 2 and 3 can be written in four different ways:
b
nd 2 to 3 2 out of 3 2:3 (cid:1)2(cid:1)
3
a
H
Rates

h
at A rate is a ratio that compares two quantities with different measurement units.
M
A unit rate is a rate that has been simplified so that the denominator is 1.
Example: Write 98 km in 2.0 hours as a unit rate.
98km
98 km in 2.0 hours is a ratio of (cid:1)(cid:1)
2.0hours
(cid:1)98(cid:1)km (cid:2)((cid:1)98(cid:1))((cid:1)km(cid:1)) Split the fraction into the product of a number fraction
2.0hours 2.0 hour and a unit fraction
(cid:2)(49)((cid:1)km(cid:1)) Simplify the number fraction
hour
(cid:2)49 km per hour or 49 km/h
Example: Write 16 Swedish crowns in 2 U.S. dollars as a unit rate.
16Swedishcrowns
16 Swedish crowns in $2.00 American currency is a ratio of (cid:1)(cid:1)(cid:1)
2U.S.dollars
1(cid:1)6Swe(cid:1)dishc(cid:1)rowns (cid:2)((cid:1)16(cid:1))((cid:1)Swedish(cid:1)crowns)
2U.S.dollars 2 U.S.dollars
(Swedishcrowns)
= (8) (cid:1)(cid:1)
U.S.dollars
= 8 Swedish crowns per U.S. dollar
or 8 Swedish crowns/U.S. dollar
Proportions
A proportion is an equation that states that two ratios are equal:
(cid:1)a(cid:1)(cid:2)(cid:1)c(cid:1), where b and d are not zero.
b d
Proportions used to solve ratio problems often include three numbers and one vari-
able. You can solve the proportion to find the value of the variable. To solve a proportion,
use cross multiplication.
Example: Solve the proportion (cid:1)a(cid:1)(cid:2)(cid:1)c(cid:1) for a.
b d
Cross multiply: (cid:1)a(cid:1)(cid:2)(cid:1)c(cid:1)
b d
ad(cid:2)bc Write the equation resulting from cross multiplying
a(cid:2)(cid:1)b(cid:1)c Solve for a
d
5. Solve the following proportions.
a. (cid:1)4(cid:1)(cid:2)(cid:1)2(cid:1) c. (cid:1)3(cid:1)6 (cid:2)(cid:1)s(cid:1)
x 3 12 16
b. (cid:1)13(cid:1)(cid:2)(cid:1)n(cid:1) d. (cid:1)2.(cid:1)5 (cid:2)(cid:1)7.(cid:1)5
15 75 5.0 w
838 Appendix A Math Handbook
833-857 EM MHB-845813 3/22/04 8:47 AM Page 839
IV. Exponents, Powers, Roots, and Absolute Value
M
Exponents
a
An exponent is a number that tells how many times a number, a, is used as a th
factor. An exponent is written as a superscript. In the term an, a is the base and n H
is the exponent. an is called the nth power of a or a raised to the nth power. an
d
b
Exponent o
o
k
an
Base
Connecting Math to Physics A subscript is not an exponent. In physics, a subscript is
used to further describe the variable. For example, v can be used to represent the velocity
0
at time 0. A subscript is a part of the variable.
Positive exponent For any nonzero number, a, and any integer, n,
an(cid:1)(a )(a )(a ). . .(a )
1 2 3 n
Example: Simplify the following exponent terms.
104(cid:1)(10)(10)(10)(10)(cid:1)10,000
23(cid:1)(2)(2)(2)(cid:1)8
Zero exponent For any nonzero number, a,
a0(cid:1)1
Examples: Simplify the following zero exponent terms.
20(cid:1)1
130(cid:1)1
Negative exponent For any nonzero number, a, and any integer, n,
a(cid:2)n(cid:1)(cid:3)1(cid:3)
an
Examples: Write the following negative exponent terms as fractions.
2(cid:2)1(cid:1)(cid:3)1(cid:3)(cid:1)(cid:3)1(cid:3) 2(cid:2)2(cid:1)(cid:3)1(cid:3)(cid:1)(cid:3)1(cid:3)
21 2 22 4
Square and Cube Roots
A square root of a number is one of its two equal factors. A radical sign, (cid:1)(cid:2),
indicates a square root. A square root can be shown as exponent (cid:3)1(cid:3), as in (cid:1)b(cid:2)(cid:1)b(cid:3)12(cid:3). You
2
can use a calculator to find square roots.
Examples: Simplify the following square root terms.
(cid:1)a(cid:2)2 (cid:1) (cid:1)((cid:2)a)(a) (cid:1)a
(cid:1)9(cid:2) (cid:1)(cid:1)((cid:2)3)(3) (cid:1)3
(cid:1)6(cid:2)4 (cid:1)(cid:1)((cid:2)8.0)(8.(cid:2)0) (cid:1) 8.0 The answer has a zero to the right of the decimal
to keep two significant digits
(cid:1)3(cid:2)8.44 (cid:1)6.200 Place two zeros to the right of the calculator answer
to keep four significant digits
(cid:1)3(cid:2)9 (cid:1)6.244997 . . .(cid:1)6.2 Round the calculator answer to keep two significant digits
Math Handbook 839
833-857 EM MHB-845813 6/8/04 10:31 PM Page 840
A cube root of a number is one of its three equal factors. A radical sign with the
k number 3, (cid:1) 3 (cid:2),indicates a cube root. A cube root also can be shown as exponent (cid:1)1(cid:1),
oo as in (cid:1)3 b(cid:2)(cid:2)b(cid:1)13(cid:1). 3
b
d
Example: Simplify the following cube root terms.
n
a
H (cid:1)3 1(cid:2)25 (cid:2)(cid:1)3 ((cid:2)5.00)(5(cid:2).00)(5.(cid:2)00)(cid:2)5.00

th (cid:1)3 3(cid:2)9.304 (cid:2)3.4000
a
M
6. Find each root. Round the answer to the nearest hundredth.
(cid:1)(cid:2) (cid:1)(cid:2)
a. 22 c. 676
b. (cid:1)3 7(cid:2)29 d. (cid:1)3 4(cid:2)6.656
7. Simplify by writing without a radical sign.
(cid:1)(cid:2)(cid:2) (cid:1)(cid:2)
a. 16a2b4 b. 9t6
8. Write using exponents.
(cid:1)(cid:2) 1
a. n3 b. (cid:1)(cid:1)
(cid:1)(cid:2)
a
Operations with Exponents
In the following operations with exponents, a and b can be numbers or variables.
Product of powers To multiply terms with the same base, add theexponents, as
in (am)(an)(cid:2)am(cid:3)n.
Quotient of powers To divide terms with the same base, subtract the bottom
exponent from the top exponent, as in am/an (cid:2)am(cid:4)n.
Power of a power To calculate the power of a power, use the same base and
multiply the exponents, as in (am)n(cid:2)amn.
nth root of a power To calculate the root of a power, use the same base and
divide the power exponent by the root exponent, as in (cid:1)n a(cid:2)m (cid:2)a(cid:1)mn(cid:1).
Power of a product To calculate the power of a product of a and b, raise both
to the power and find their product, as in (ab)n (cid:2)anbn.
9. Write an equivalent form using the properties of exponents.
a. (cid:1)x2(cid:1)t b. (cid:1)t(cid:2)3 c. (d2n)2 d. x2(cid:1)x(cid:2)
x3
(cid:3)(cid:4)
m 2qv
10. Simplify (cid:1)(cid:1) (cid:1)(cid:1)
q m
840 Appendix A Math Handbook
833-857 EM MHB-845813 3/22/04 8:48 AM Page 841
Absolute Value
The absolute value of a number, n, is its magnitude, regardless of its sign. The M
absolute value of n is written asn. Because magnitudes cannot be less than zero, a
t
h
absolute values always are greater than or equal to zero.

H
Examples: 3 units 3 units an
d
3(cid:1)3 b
o
(cid:2)3(cid:1)3 (cid:1)4 (cid:1)3 (cid:1)2 (cid:1)1 0 1 2 3 4 o
k
V. Scientific Notation
A number of the form a(cid:3)10n is written in scientific notation, where 1(cid:4)a(cid:4)10, and n
is an integer. The base, 10, is raised to a power, n. The term a must be less than 10.
Power
(cid:2) n
a 10
Term Base 10
Connecting Math to Physics Physicists use scientific notation to express, compare,
and calculate with measurements that are greater than 10 or less than 1. For example,
the mass of a proton is written as 6.73(cid:3)10(cid:2)28kg. The density of water is written as
1.000(cid:3)103kg/m3. This shows, using significant digit rules, that this measurement is
exactly 1000 to four significant digits. However, writing the density of water as 1000
kg/m3 would imply that it has only one significant digit, which is incorrect. Scientific
notation helps physicists keep accurate track of significant digits.
Large Numbers—Using Positive Exponents
Multiplying by a power of 10 is like moving the decimal point that same number of
places to the left (if the power is negative) or right (if the power is positive). To express a
large number in scientific notation, first determine the value for a, 1(cid:4)a(cid:5)10. Count the
number of decimal places from the decimal point in a to the decimal point in the number.
Use that count as the power of 10. A calculator shows scientific notation with e for
exponent, as in 2.4e(cid:6)11(cid:1)2.4(cid:3)1011. Some calculators use an E to show the exponent,
or there is often a place on the display where the calculator can show smaller-sized
digits representing the exponent.
Example: Write 7,530,000 in scientific notation.
The value for a is 7.53. (The decimal point is to the right of the first nonzero digit.)
So the form will be 7.53(cid:3)10n.
7,530,000(cid:1)7.53(cid:3)106 There are six decimal places, so the power is 6
To write the standard form of a number expressed in scientific notation, write the
value of a,and place extra zeros to the right of the number. Use the power and move the
decimal point in a that many places to the right.
Example: Write the following number in standard form.
2.389(cid:3)105(cid:1)2.38900(cid:3)105(cid:1)238,900
Math Handbook 841