Chapter 10 Physics & Math: Mathematics

hecto

h, 10²

kilo

k, 10³

logA A

1

1 electron volt to joule

1.602x10^-19 Joule

1 amu to kg

1.661x10^-27 kg

common logarithms

base-ten logs, log₁₀

centi

c, 10⁻²

division with exponents

in division, if both bases are the same, subtract the exponent of the denominator from the exponent of the numerator, e.g. X⁹/ X³= X⁶

√2

~1.4

cos 180

-1

log 1/A

-logA

logA 1

0

X⁻⁵

1/X⁵

1 cal to joule

4.184J

1lb to newton

4.45N

20²

400

1 mile to feet

5280 ft

giga

G, 10⁹

conversion between natural and common logs

log x= lnx/2.303

significant figures

provide an indication of our certainty of measurement when performing calculatinons. Determined by the precision of the instrument being used for measurement.

estimate value of 15.4 / 3.80

shift both numbers in the same direction, adjust the divisor (number you are dividing by) first. 16/4~ 4

(X/Y)⁴

X⁴/Y⁴

√3

~1.7

e

~2.718

X^ (2/3)

³√X²

sin 180

0

sin0

0

tan 180

0

mega

M, 10⁶

tetra

T, 10¹²

tan 90

undefined

tan0

0

X⁰

1

cos0

1

how to determine number of significant figures

1. count all numbers between first nonzero digit on left and last nonzero digit on right. Any digit in between is significant 2. any zeros to the left of the first nonzero digit are leading zeros, not significant 3. if there are zeros to the right of the last nonzero digit and there is a decimal point, they are significant 4. for measurements, last number is usually an approximation and is not significant

substitution for algebraic systems 5x-2y=11 3x+4y=17

1. solve for one variable, y=5x-11/2 2. insert expression into other equation, 3x+4(5x-11/2)=17 3. isolate the variable and solve resulting equation, x=3 4. solve for the other variable using that, y=2

algebraic systems equations by setting equations equal 5x-2y=11 3x+4y=17

1. solve for the same variable on both sides of the equation and then set the two equations equal to each other y=5x-11/2 and y=17-3x/4 2. set equations equal, isolate variable and solve for it, x=3 3. solve the other variable by plugging in, y=2

11²

121

12²

144

13²

169

14²

196

1 inch to cm

2.54 cm

15²

225

16²

256

17²

289

18²

324

1L to ounces

33.8 ounces

19²

361

log A^B

BlogA

determine cylinder volume with radius that is measured 7.45m and height 8.323m

V=A(base)* height. Radius has least number of significant digits, 2, so answer is 1.5*10^3

inverse relationships

an increase in one variable is associated with a proportional decrease in the other

speedometer registers at 35mph. What is the speed in meters per second?

convert distance measurements with dimensional analysis to get 56,327m/hr convert time measurements to get 15.6m/s

estimating logs

convert the log(number) to scientific notation. Log (nx10^m)=log(n)+log(10^m)= m+log(n). n is between 1-10, so its log will be a decimal between 0-10. The closer n is to 1, the closer log n will be to 0; the closer n is to 10, the closer logn will be to 1. Approximate that log (nx10^m) is m+0.n, e.g. log (9.2x10^8) is 8>0.92=8.92

deci

d, 10⁻¹

deka

da, 10¹

ejection fraction is proportional to left ventricular volume expelled with each contraction of heart. Patient has ejection fraction of .7, cardiac output is 5L/min, HR is 80bpm. What is volume of left ventricle in this person?

determine volume ejected per beat: 5L/min / 80bpm= .0625L/beat. Only 70% of volume is expelled per heart beat, so we can determine the volume expelled: .0625L/beat x .7/beat= .0625/.7= .0893L

solving square roots that are not perfect

divide a number into known squares in attempt to reduce it, e.g. √180= √4x √9x √5= 2x3x√5= 6√5. Then estimate by knowing that it is between 2 and 3 because those squares are 4 and 9. Estimate √5=2.2, then 6(2.2)=13.2

inverse trignometric fucntions

each trigonometric function also has an inverse function, e.g. sin⁻¹. Can use length measurements to find an angle, e.g. sin⁻¹(a/c)= angle

multiplication with exponents

exponents can be manipulated directly if the base number is the same. When multiplying two numbers with the same base, the exponents are added to determine the new number, e.g. X³+X⁶=X⁹

solving division with scientific notation

extend out to get a perfect square. Since radical is 1/2 exponent, multiply the exponent inside by 1/2, e.g. √(4.9x10^-7) = √49x10^-8=7x10^-4

(X³)⁴

for a number that is raised to an exponent and then raised again to another exponent, the two exponents are multiplied, X¹²

estimation in multiplication

for complex multiplication problems, e.g. (3.17x 10^4) x (4.53x10^5), round the decimal place back. If you round one answer up, round the other answer down to compensate. Estimation would be (3.2x10^4) x (4.5x10^5)

math with significant figures

for multiplication and division, maintain as many digits as possible, then round to have as many significant digits that is the same as the number with least amount of significant digits. For addition and subtraction, decimal points are maintained. At one end, leave as many numbers after decimal point as appear in the number with least numbers after decimal

direct relationships

increasing one variable proportionately increases the other; as one decreases, the other decreases by the same proprotion

log AxB

logA + logB

log A/B

logA - logB

natural logs

loge or ln

milli

m, 10⁻³

estimation in division

make proportional adjustments in the same direction, e.g. round both numbers up or both numbers down

scientific notation

method of writing numbers that takes advantage of a powers of ten. A number is written with a significand and an exponent

elimination to solve algebraic systems of equation 5x-2y=11 3x+4y=17

multiply or divide one (or both) of the equations to get the same coefficient in front of one of the variables in both equations. Then add or subtract equations as necessary to eliminate one of the variables. Solve for one variable, then use it to solve for the other variable

nano

n, 10⁻⁹

p

p can be shorthand for -log, e.g. pH= -log[H+]

pico

p, 10⁻¹²

significand

the number before the multiplication sign in scientific notation. Has an absolute value in the range [0,10). Cannot begin with 0 or have >1 digit before the decimal place

micro

u, 10⁻⁶

adding numbers with exponents

when adding or subtracting numbers with exponents, true value must be calculated before addition or subtraction can be performed, e.g. 3^2 + 3^2= 18, not 6^2