Transcript MATH PROBLEMS TOPIC ONE

MATH PROBLEMS
TOPIC ONE
Measurement, Uncertainty,
Significant Figures
The age of the universe is thought to be
about 14 billion years. Assuming 2 significant
figures, write this in powers of ten in
a) Years
b) Seconds
ANSWER
A) 14 Billion years = 14,000,000,000 years

1.4 x 1010 years
B)
365days 24hours 60min 60sec 
17
1.4
x10
years

4.4
x10
sec





 1year  1day 1hour 1min 
10
QUESTION
How many significant figures do each of the
following numbers have:
A.
214
E.
0.0086
B.
81.60
F.
3236
C.
7.03
G.
8700
D.
0.03
ANSWER
.
A.
214
3 Significant Figures
B.
81.60
4 Significant Figures
C.
7.03
3 Significant Figures
D.
0.03
1 Significant Figure
E.
0.0086
2 Significant Figures
F.
3236
4 Significant Figures
G.
8700
2 Significant Figures
QUESTION
Write the following numbers in powers of ten
notation (Scientific Notation):
A.
1.156
B.
21.8
C.
0.0068
D.
27.635
E.
0.219
F.
444
ANSWER
A.
1.156
 1.1.56 x 100
B.
21.8
 2.18 x 101
C.
0.0068
 6.8 x 10-3
D.
27.635
 2.7635 x 101
E.
0.219
 2.19 x 10-1
F.
444
 4.44 x 102
QUESTION
An electron can tunnel through an energy
barrier with probability 0.0000000000375.
(This is a concept used in quantum
mechanics.) Express this probability in
scientific notation.
* Answer:
3.75 x 10−11
QUESTION
Sara has lived 18.0 years. How many seconds
has she lived? Express the answer in scientific
notation. Use 365.24 days per year for your
calculations.
* Answer:
5.68 x 108 s
365.24days 24hours 60min 60sec 
8
18years

5.68x10
sec






 1year  1day 1hour 1min 
QUESTION
Write out the following numbers in full with
the correct number of zeros:
A.
8.69 x 104
B.
9.1 x 103
C.
8.8 x 10-1
D.
4.76 x 102
E.
3.62 x 10-5
ANSWER
A.
8.69 x 104
86,900
B.
9.1 x 103
9,100
C.
8.8 x 10-1
0.88
D.
4.76 x 102
476
E.
3.62 x 10-5
0.0000362
QUESTION
Express the following sum with the correct number of
significant figures:
1.80 m + 142.5 cm + 5.34 x 105 μm
* To add values with significant figures, adjust all values to be
added so that their units are all the same
1.80 m + 142.5 cm + 5.35 x 105 μm =
1.80 m + 1.425 m + 0.534 m = 3.759 m  3.76 m
When adding, the final result is to be no more accurate than
the least accurate number used. In this case, that is the first
measurement, which is accurate to the hundredths place.
*
QUESTION
What is the percent uncertainty in the
measurement 3.76 +/- 0.25 m?
PERCENT UNCERTAINTY:
Percent Uncertainty = Uncertainty /Measurement x 100%
* Measurement = 3.76 m
* Uncertainty = 0.25 m
0.25 m
x100%  6.6%


3.76 m
QUESTION
Time intervals measured with a stopwatch
typically have an uncertainty of about 0.2s,
due to human reaction time at the start and
stop moments. What is the percent
uncertainty of a handtimed measurement of
A) 5 s
B) 50 s
C) 5 min
ANSWER
Percent uncertainty = Uncertainty / Measurement x 100%
A)
% Uncertainty =
0.2 sec 
x100%  4%


 5 sec 
% Uncertainty =
0.2 sec 
x100%  0.4%


50 sec 
B)

ANSWER
Percent uncertainty = Uncertainty / Measurement x 100%
C)
% Uncertainty =

0.2 sec 
x100%  0.07%


300 sec 
QUESTION
Express each of the following numbers to only
three significant figures:
A.
10.072 m
B.
775.4 km
C.
0.002549 kg
D.
93,000,000 mi
ANSWER
Express each of the following numbers to only
three significant figures:
A.
10.072 m
10.1 m
B.
775.4 km
775 km
C.
0.002549 kg
2.55 x 10-3 kg
D.
93,000,000 mi
9.30 x 107 mi
QUESTION
* Multiplying Numbers in Scientific Notation
Multiply 3.65×1023 by 4.12×10154 by 1.11×10−11 and express
the answer in scientific notation.
*ANSWER: 1.67 x 10167
1st:
Add together exponents
(23+154+(-11)) = 166
2nd:
Multiply (3.65)(4.12)(1.11) = 16.7
16.7  1.67 x 101  1+166 = 167
**ANSWER  1.67 x 10167
QUESTION
*Dividing Numbers in Scientific Notation
Newton's second law states that the net force
equals the product of mass and acceleration.
A boat's mass of is 9.6×105 kg and it
experiences a net force of 1.5×104 kg∙m/s2.
State its acceleration.
* GIVEN  F = ma
ANSWER
*Dividing Numbers in Scientific Notation
Boat's mass = 9.6×105 kg
Net force = 1.5×104 kg∙m/s2
* GIVEN  F = ma
* Acceleration = ?
ANSWER:
1.6 x 10−2 m/s2
1.5x10 4 kgxm
5
2
2

/(9.6x10 kg)  1.6x10 m /s
2
sec


ANSWER
*Dividing Numbers in Scientific Notation
Boat's mass = 9.6×105 kg
kg∙m/s2
Net force = 1.5×104
1.5×104 kg∙m/s2 / 9.6×105 kg
1st: Subtract the exponents
4 – 5 = -1
2nd: Divide 1.5/9.6 = 0.16  1.6 x 10-1
3rd: Add together the exponents
(-1) + (-1) = -2
** ANSWER: 1.6 x 10-2 m/s2
QUESTION
* Adding and Subtracting Numbers in
Scientific Notation:
A 3.70×106 kg piece splits off an iceberg of mass of
5.96×107 kg. Calculate the mass of the remaining
iceberg and express the answer in scientific notation.
(5.96 × 107 kg) – (3.70 x 106 kg) =
(5.96 × 107 kg) – (0.370 x 107 kg)=
(5.96 – 0.370) × 107 kg =
5.59 x 107 kg
* ANSWER:
5.59 x 107 kg
QUESTION
* Adding and Subtracting Numbers in Scientific
Notation:
A small metal chamber contains 5.632×10−9 mol of oxygen gas
at very low pressure. An additional 4.379×10−8 mol of oxygen
gas are added. Find the total amount of gas in the chamber.
(5.632 × 10−9 mol) + (4.379 × 10−8 mol) =
(5.632 × 10−9 mol) + (4.379 × 10−8 mol) =
(0.5632 × 10−8 mol) + (4.379 × 10−8 mol) =
(0.5632 + 4.379) × 10−8 mol =
* ANSWER:
4.942 x 10−8 mol
Standard Units,
Converting Units
Which of the following is not an SI base
quantity?
A) Length
B) Mass
C) Weight
D) Time
Standard Units,
Converting Units
Which of the following is not an SI base
quantity?
A) Length
B) Mass
C) Weight
D) Time
QUESTION
Which of the following is the SI unit of mass?
a) pound
b) gram
c) kilogram
d) ton
ANSWER
Which of the following is the SI unit of mass?
a) pound
b) gram
c) kilogram
d) ton
QUESTION
How many centimeters are there in a
kilometer?
*Answer:
100,000 cm
1000m 100cm 

100,000cm
1km
 1km  1m 
QUESTION
Write the following as full (decimal) numbers
with standard units:
a) 286.6 mm
b) 85 μV
c) 760 mg
d) 60.0 ps
e) 2.50 gigavolts
ANSWER
a) 286.6 mm  Standard Unit = Meter
1000 mm = 1 m
 1m 
 0.2866m
286.6mm
1000mm 
b) 85 μV  Standard Unit = Volt
1000 microvolt (μV) = 1 millivolt
 = 1 volt
1000 millivolt
 1mV  1V 
 0.000085V
85m

1000m 1000mV 
ANSWER
c) 760 mg  Standard unit = kg
1000 mg = 1 g
1000 g = 1 kg
 1g  1kg 
760mg

 0.000760kg
1000mg 1000g 

ANSWER
d) 60.0 picoseconds  Standard unit =
Second
1 x 10-12 s = 1 ps or 1 x 1012 ps = 1 s
 1s 
60.0 ps 12  0.0000000000600s
1x10 ps 
ANSWER
e) 2.50 gigavolts  Standard unit = Volts
1 x 109 volts = 1 gigavolt
 1x10 9 V 
2.50gigavolts
 2,500,000,000V
1gigavolt 
QUESTION
A drug company has just manufactured 50.0
kg of acetylsalicylic acid for use in aspirin
tablets. If a single tablet contains 500 mg of
the drug, how many tablets can the company
make out of this batch?
ANSWER
* ANSWER:
100,000 tablets
1000g 1000mg  1tablet 
50.0kg


 100,000tablets
 1kg  1g 500mgdrug 
QUESTION
Determine your own height in meters, and
your mass in kilograms.
* Example- Height = 5’8” (5 feet 8 inches)
* Given: 1 meter = 39.37 inches
 1meter 
 1.7meters
68inches
39.37inches 
QUESTION
Determine your own height in meters, and your
mass in kilograms.
* Example- Weight = 165 lbs
* Given: 0.456 kg = 1 lb
0.456kg
 75.2kg
165lbs
 1lb 
* Technically, pounds and mass measure 2 separate properties. To
make this conversion, we have to assume that we are at a
location where the acceleration due to gravity is 9.8 m/s2
QUESTION
The Sun, on average, is 93 million miles from
Earth. How many meters is this? Express
a) Powers of ten notation (Scientific
Notation)
b) Using a metric prefix
ANSWER
a) Express using powers of ten…
1 mile = 1610 meters
93 million miles = (93 x 106 miles)
1610meters
11
93x10
miles

1.5x10
meters

 1mile 
6
ANSWER
b) Express using a metric prefix…
1.5 x 1011 m  150 x 109 m = 150 gigameters
OR
1.5 x 1011 m  0.15 x 1012 m = 0.15 terameters
QUESTION
What is the conversion factor between…
a) ft2 and yd2
 
2 1yd
2
1
ft

0.111yd
 3 ft 
2
1 yard = 3 feet
b) m2 and ft2
1 meter = 3.28
 feet

2


3.28 ft
2
2
1m

10.8
ft
  1m 
QUESTION
An airplane travels at 950 km/h. How long does
it take to travel 1.00 km?
* Use the speed if the airplane to convert the
travel distance into a time.
1hour 60min 60sec 


 3.8sec
1.00km
950km1hour 1min 
QUESTION
A typical atom has a diameter of about 1.0 x 10-10
m.
A) What is this in inches?
Given: 1 m = 3.28 ft
1.0x10
10
3.28 ft 12inches 
9
m

 3.9x10 in
 1m  1 ft 
QUESTION
A typical atom has a diameter of about 1.0 x 10-10
m.
B) Approximately how many atoms are there
along a 1.0-cm line?
 1m  1atom 
8
1.0cm
1.0x10
atoms




10
100cm 1x10 m 
QUESTION
11.2 meters per second is how many miles
per hour?
* ANSWER:
25.0 mi/h
11.2meters60sec 3.28 feet  1mile 




 25mi /h
 1sec 1hour  1meter 5280 feet 
QUESTION
A light-year is the distance light travels in one
year (at speed = 2.998 x 108 m/s).
A) How many meters are there in one light
year?
Know  1 light year = 2.998 x 108 m/s
* How many seconds are in one year?
365days 24hours 60min 60sec 
7

 3.156x10 sec/ year



 1year  1day 1hour 1min 
ANSWER
A light-year is the distance light travels in one
year (at speed = 2.998 x 108 m/s).
A) How many meters are there in one light year?
Know  1 light year = 2.998 x 108 m/s
3.156x10 7 sec 2.998x10 8 meters
15


 9.46x10 meters/ year
1sec
 1year


*Answer = 9.46 x 1015 meters

QUESTION
A light-year is the distance light travels in one
year (at speed = 2.998 x 108 m/s).
B) An astronomical unit (AU) is the average
distance from the Sun to Earth, 1.50 x 108 km.
How many AU are there in 1.00 light-year?
Know 
1 light year = 9.46 x 1015 meters
1 AU = 1.50 x 108 km
9.46x1015 meters

1AU
4

6.31x10
AU /lightyear



11
 1lightyear 1.50x10 meters
QUESTION
A light-year is the distance light travels in one
year (at speed = 2.998 x 108 m/s).
C) What is the speed of light in AU/h?
Know 
1 light year = 6.31 x 104 AU
1 AU = 1.50 x 1011 m
1 light year = 2.998 x 108 m/s
2.998x10 8 meters
3600sec 
1AU

 7.20AU /hr


11
1sec

1.50x10 meters 1hr 
Dimensional Analysis
The following variables are commonly seen in
equations. The name of the quantity represented by
each variable, and its dimension(s), are also shown.
x distance (L)
a acceleration (L/T2)
t time (T)
v speed (L/T)
m mass (M)
F force (ML/T2)
Using the information above, select the equations that are
dimensionally correct.
F = ma
v2 = 2ax
v = at2
F/v = m/t
Dimensional Analysis
Answer
F = ma
v2 = 2ax
F/v = m/t
QUESTION
The dimensions for force are the product of mass
and length divided by time squared. Newton's
second law states that force equals the product of
mass and acceleration. What are the dimensions
of acceleration?
A)
T2
B)
T2/L
C)
L/T2
D)
L/T
QUESTION
The dimensions for force are the product of mass
and length divided by time squared. Newton's
second law states that force equals the product of
mass and acceleration. What are the dimensions
of acceleration?
A)
T2
B)
T2/L
C)
L/T2
D)
L/T
* Disclaimer: This powerpoint presentation is a compilation of various works.