Transcript Uncertainty in Measurements

Errors and uncertainties in chemistry
internal assessment
• The consideration and appreciation of the significance of
the concepts of errors and uncertainties helps to develop
skills of inquiry and thinking that are not only relevant to
the group 4 experimental sciences
• The treatment of errors and uncertainties is directly
relevant in the internal assessment criteria of:
– data collection and processing, aspects 1 and 3
(recording raw data and presenting processed data)
– conclusion and evaluation, aspects 1, 2 and 3
(concluding, evaluating procedure(s), and improving
the investigation).
Within internal assessment students should be
able to do the following:
• make a quantitative record of uncertainty range
(±) (data collection and processing: aspect 1)
• state the results of calculations to the
appropriate number of significant figures. The
number of significant figures in any answer
should reflect the number of significant figures in
the given data (data collection and processing:
aspect 3).
• propagate uncertainties through a calculation so
as to determine the uncertainties in calculated
results and to state them as absolute and/or
percentage uncertainties.
Random and systematic errors
• Systematic errors arise from a problem in
the experimental set-up that results in the
measured values always deviating from
the “true” value in the same direction, that
is, always higher or always lower.
Examples of causes of systematic error
are miscalibration of a measuring device
or poor insulation in calorimetry
experiments.
• Random errors arise from the imprecision
of measurements and can lead to readings
being above or below the “true” value.
Random errors can be reduced with the
use of more precise measuring equipment
or its effect minimized through repeat
measurements so that the random errors
cancel out.
Uncertainties in raw data
• When numerical data is collected, values cannot
be determined exactly, regardless of the nature
of the scale or the instrument. If the mass of an
object is determined with a digital balance
reading to 0.1 g, the actual value lies in a range
above and below the reading. This range is the
uncertainty of the measurement. If the same
object is measured on a balance reading to
0.001 g, the uncertainty is reduced, but it can
never be completely eliminated. When recording
raw data, estimated uncertainties should be
indicated for all measurements.
Uncertainty in Measurements
Measurements always involve a comparison. When you say that a
table is 6 feet long, you're really saying that the table is six times longer
than an object that is 1 foot long. The foot is a unit; you measure the
length of the table by comparing it with an object like a yardstick or a tape
measure that is a known number of feet long.
The comparison always involves some uncertainty. If the tape
measure has marks every foot, and the table falls between the sixth and
seventh marks, you can be certain that the table is longer than six feet
and less than seven feet. To get a better idea of how long the table
actually is , though, you will have to read between the scale division
marks. This is done by estimating the measurement to the nearest one
tenth of the space between scale divisions.
Which of the following best describes the length of the beetle's
body in the picture to the left? Between 0 and 2 in
Between 1 and 2 in
Between 1.5 and 1.6 in
Between 1.54 and 1.56 in
Between 1.546 and 1.547 in
Uncertainty in Length Measurements
• Measurements are often written as a single number
rather than a range. The beetle's length in the previous
frame was between 1.54 and 1.56 inches long. The
single number that best represents the measurement is
the center of the range, 1.55 inches. When you write the
measurement as a single number, it's understood that
the last figure (the 5 in this case) had to be estimated.
Consider measuring the length of the same object with
two different rulers.
• Give the correct length measurement for the steel pellet
for each of the rulers, as a single number rather than a
range:
Left Ruler: ................ in
Right Ruler: .............. in
ANSWERS
• The left ruler has scale markings every inch,
so you must estimate the length of the pellet
to the nearest 1/10 of an inch. 1.4 in, 1.5 in, or
1.6 in would be acceptable answers.
• The right ruler has scale markings every 0.1
inches, so you must estimate the length of
the pellet to the nearest 0.01 inches. 1.46
inches is an acceptable answer.
Uncertainty in Length Measurements
• Give the correct length measurement for this
electronic component for each of the rulers,
as a single number rather than a range:
Blue Ruler:................ cm
White Ruler:.............. cm
The blue ruler has scale markings every
0.1 cm, so you must estimate the length
of the electronic component to the
nearest 0.01 cm. 1.85 cm, 1.86 cm, or 1.87
cm would be acceptable answers.
The white ruler has scale markings every 1 cm, so you must estimate the
length of the electronic component to the nearest 0.1 cm. 1.9 cm is an
acceptable answer.
Uncertainty in Temperature Measurements
• A zero will occur in the last
place of a measurement if
the the measured value fell
exactly on a scale division.
For example, the
temperature on the
thermometer just should be
recorded as 30.0°C.
Reporting the temperature
as 30°C would imply that the
measurement had been
taken on a thermometer with
scale marks 10°C apart!
Uncertainty in Temperature Measurements
• A temperature of 17.00°C
was recorded with one of
the three thermometers to
the left. Which one was it?
the top one
the middle one
the bottom one
either the top one, or the
middle one
either the middle one, or the
bottom one
it could have been any of
them
ANSWERS
• The top thermometer had scale markings
0.1°C apart; it could be read to the nearest
0.01°C.
• The middle thermometer has scale markings
ever 0.2°C, so it can be read to the nearest
0.02°C.
• The bottom thermometer has markings every
degree, and can be read to the nearest tenth
of a degree. Try again.
Uncertainty in Volume Measurements
• Use the bottom of the meniscus (the
curved interface between air and
liquid) as a point of reference in
making measurements of volume in
a graduated cylinder, pipet, or buret.
In reading any scale, your line of
sight should be perpendicular to the
scale to avoid 'parallax' reading
errors.
• The graduated cylinder on the left
has scale marks 0.1 mL apart, so it
can be read to the nearest 0.01 mL.
Reading across the bottom of the
meniscus, a reading of 5.72 mL is
reasonable (5.73 mL or 5.71 mL are
acceptable, too). Enter the volume
readings for the middle and right
cylinders below, assuming each
scale is in mL.
• Middle Cylinder Volume:................mL
• Right Cylinder Volume:................mL
ANSWERS
• The middle cylinder has graduations every mL, and can
be read to the nearest 0.1 mL. Since the meniscus
touches the mark, the reading should be recorded as 3.0
mL, NOT as 3 mL. If you read 3.1 mL, you were probably
reading across the top of the meniscus. Read at the
bottom of the meniscus.
• The right cylinder has graduations every 0.1 mL, and
can be read to the nearest 0.01 mL. Since the meniscus
is just below the halfway mark between 0.3 and 0.4, the
reading should be recorded as 0.34 mL (although
readings of 0.35 mL or 0.33 mL are acceptable). If you
read 0.37 or 0.38, you were probably reading across the
top of the meniscus. Read at the bottom of the
meniscus.
C. Estimating the uncertainity ( propogation of
errors )
Device
Example
Uncertainity
Analogue scale
Ruler,voltmeter,ammeter
,graduated cylinder, thermometer,
watch, stopwatch,meters with
moving pointers.
half of the smallest scale
division
Top-pan balances digital
meters,voltmeter,pH meter
the smallest scale division

Digital scale
Reflex time of a person

0.2 seconds
Example
1. If smallest division of a ruler is 1 mm, than uncertainity range is 0.5 mm= 0.05 cm = 0.0005 m
If smallest division of graduated cylinder is 5 cm3 , than uncertainity range is 2.5cm3.
If smallest division of graduated cylinder is 10 cm3 , than uncertainity range is 5cm3.
If smallest division of graduated cylinder is 1 cm3 , than uncertainity range is 0.5cm3.
If smallest division of graduated cylinder is 1 mm3 , than uncertainity range is 0.5mm3.
Propagating errors
• Random errors (uncertainties) in raw data
feed through a calculation to give an error
in the final calculated result. There is a
range of protocols for propagating errors.
A simple protocol is as follows:
•
When adding or subtracting quantities, then
the absolute uncertainties are added.
For example, if the initial and final burette
readings in a titration each have an uncertainty
of ±0.05 cm3 then the propagated uncertainty
for the total volume is (±0.05 cm3) +
(±0.05 cm3) = (±0.10 cm3).
•
When multiplying or dividing
quantities, then the percent (or
fractional) uncertainties are added.
For example:
molarity of NaOH(aq) =
1.00 M (±0.05 M)
percent uncertainty =
[0.05/1.00]×100 = 5%
volume of NaOH(aq) =
10.00 cm3 (±0.10 cm3)
percent uncertainty =
[0.10/10.00]×100 = 1%
• Therefore, calculated moles of NaOH in
solution = 1.00×[10.00/1000] = 0.0100
moles (±6%)
• The student may convert the calculated
total percent uncertainty back into an
absolute error or leave it as a percentage.
Exact Numbers
•
•
•
•
Numbers obtained by counting have no uncertainty unless the count is very
large. For example, the word 'sesquipedalian' has 14 letters. "14 letters" is
not a measurement, since that would imply that we were uncertain about
the count in the ones place. 14 is an exact number here.
Very large counts often do have some uncertainty in them, because of
inherent flaws in the counting process or because the count fluctuates. For
example, the number of human beings in the state of Maryland would be
considered a measurement because it can not be determined exactly at the
present time.
Numbers obtained from definitions have no uncertainty unless they have
been rounded off. For example, a foot is exactly 12 inches. The 12 is not
uncertain at all. A foot is also exactly 30.48 centimeters from the definition
of the centimeter. The 8 in 30.48 is not uncertain at all. But if you say 1 foot
is 30.5 centimeters, you've rounded off the definition and the rounded digit
is uncertain.
Which of the following quantities can be determined exactly? (Select all that
are NOT measurements.)
The number of light switches in the room you're sitting in now
The number of ounces in one pound
The number of stars in the sky
The number of inches per meter
The number of red blood cells in exactly one quart of blood
ANSWERS
• Anything that can be easily counted is exact.
The number of light switches in the room
you're sitting in now is exact, for
example.Any defined quantity is exact. The
number of ounces in one pound is exactly
16. The number of inches per meter must be
exact since there are exactly 30.48
centimeters in a foot, exactly 12 inches in a
foot, and exactly 100 centimeters in a meter.
• Stars in the sky and red blood cells in a
given volume of blood can be counted, but
the counts are so large that there will
inevitably be some uncertainty in the final
result.
What are Significant Digits?
All of the digits up to and including the estimated digit are
called significant digits. Consider the following
measurements. The estimated digit is red:
Measurement
142.7 g
103 nm
2.99798 x 108 m
Number of
Significant
Digits
4
3
6
Distance between Markings
onMeasuring Device
1g
10 nm
0.0001 x 108 m
Significant Figures
• The significant figures of a (measured or
calculated) quantity are the meaningful
digits in it. There are conventions which
you should learn and follow for how to
express numbers so as to properly
indicate their significant figures
Any digit that is not zero is significant. Thus 549 has three
significant figures and 1.892 has four significant figures.
Zeros between non zero digits are significant. Thus 4023 has
four significant figures.
Zeros to the left of the first non zero digit are not significant.
Thus 0.000034 has only two significant figures. This is more
easily seen if it is written as 3.4x10-5.
For numbers with decimal points, zeros to the right of a non
zero digit are significant. Thus 2.00 has three significant figures
and 0.050 has two significant figures. For this reason it is
important to keep the trailing zeros to indicate the actual number
of significant figures.
•For numbers without decimal points, trailing zeros may or may not be
significant. Thus, 400 indicates only one significant figure. To indicate
that the trailing zeros are significant a decimal point must be added. For
example, 400. has three significant figures, and 4 x10^2 has one
significant figure.
•Exact numbers have an infinite number of significant digits. For
example, if there are two oranges on a table, then the number of
oranges is 2.000... . Defined numbers are also like this. For example,
the number of centimeters per inch (2.54) has an infinite number of
significant digits, as does the speed of light (299792458 m/s)
There are also specific rules for how to consistently
express the uncertainty associated with a number. In
general, the last significant figure in any result should
be of the same order of magnitude (i.e.. in the same
decimal position) as the uncertainty. Also, the
uncertainty should be rounded to one or two significant
figures. Always work out the uncertainty after finding
the number of significant figures for the actual
measurement.
For example,
9.82 +/- 0.02
10.0 +/- 1.5
4 +/- 1
QUESTIONS
A sample of liquid has a measured volume of 23.01 mL.
Assume that the measurement was recorded properly.
1. How many significant digits does the
measurement have?
Enter your answer here:…………..
2. Suppose the volume measurement was made with
a graduated cylinder. How far apart were the scale
divisions on the cylinder, in mL?
A) 10 mL B)1 mL C) 0.1 mL D) 0.01 mL
E) not enough information
3. Which of the digits in the measurement is
uncertain?
A) "2" B) "3" C)"0" D)"1"
E)not enough information
ANSWERS
1. This measurement had FOUR significant figures.
The 2, 3, and 0 are certain; the 1 is uncertain.
Significant digits include all of the figures up to and
including the first uncertain digit.
2. Since the hundredths place was uncertain, the
graduated cylinder must have had markings 0.1 mL
apart.
3. The last figure is the uncertain one; writing the
number as 23.01 mL means: My best estimate of the
volume is 23.01 mL, but it could have been 23.00 mL
or maybe 23.02 mL.
QUESTIONS
A piece of steel has a measured mass of 1.0278 g.
Assume that the measurement w as recorded properly.
1. How many significant digits does the
measurement have?
Enter your answer here: ………….
2. How far apart were the marks on the scale the
mass was read from, in g?
A) 1 g B) 0.1 g C) 0.01 g D) 0.001 g E) 0.0001 g
F) 0.00001 g G) not enough information
3. Which of the digits in the measurement is
uncertain?
A) "1" B) "0" C) "2" D) "7" E) "8"
F) not enough information
ANSWERS
1. This measurement had FIVE significant
figures. The 1, 0, 2, and 7 are certain; the 8 is
uncertain. Significant digits include all of the
figures up to and including the first uncertain
digit.
2. Since the ten thousandths place was
uncertain, the scale must have had markings
0.001 g (1 mg) apart.
3. The last figure is the uncertain one;
writing the number as 1.0278 g means: My
best estimate of the volume is 1.0278 g, but it
could have been 1.0277 g or maybe 1.0279.
Counting Significant Digits
Moving the Decimal Point Doesn't Change
Significant Figures
Usually one can count significant digits simply by
counting all of the digits up to and including the
estimated digit. It's important to realize, however, that
the position of the decimal point has nothing to do with
the number of significant digits in a measurement. For
example, you can write a mass measured as 124.1 g as
0.1241 kg. Moving the decimal place doesn't change the
fact that this measurement has FOUR significant figures.
Suppose a mass is given as 127 ng. That's 0.127 µg, or
0.000127 mg, or 0.000000127 g. These are all just
different ways of writing the same measurement, and all
have the same number of significant digits: three.
Counting Significant Digits
Moving the Decimal Point Doesn't Change
Significant Figures
If significant digits are all digits up to and including the
first estimated digit, why don't those zeros count? If
they did, you could change the amount of uncertainty in
a measurement that significant figures imply simply by
changing the units. If 0.00125 L has 3 significant digits,
you know the uncertainty is about 1 part in 100 (1 part
in 125, to be exact). If it had 6 significant digits, the
uncertainty would only be about 1 part in a million. By
not counting those leading zeros, you ensure that the
measurement has the same number of figures (and the
same relative amount of uncertainty) whether you write
it as 0.00125 L, 1.25 mL, or 1250 µL.
QUESTIONS
Fill in the blanks in the following table.
Measurement
Number of
Significant
Figures
1. 0.000341 kg = 0.341 g = 341 mg
2. 12 mg = 0.000012 g = 0.000000012 kg
3. 0.01061 Mg = 10.61 kg = 10610 g
………………
………………
………………
ANSWERS
1. 0.000341 kg = 0.341 g = 341 mg. 341 mg
clearly has 3 significant figures, and so must
the same measurement written in kg or g.
2. 12 µg = 0.000012 g = 0.000000012 kg. 12
µg clearly has 2 significant figures, and so
must the same measurement written in g or
kg.
3. 0.01061 Mg = 10.61 kg = 10610 g. 10.61 kg
clearly has 4 significant figures, and so must
the same measurement written in g or Mg.
Counting Significant Digits
•
•
•
•
A Procedure for Counting Significant Digits
How can you avoid counting zeros that serve
merely to locate the decimal point as
significant figures? Follow this simple
procedure:
Move the decimal point so that it is just to
the right of the first nonzero digit, as you
would in converting the number to scientific
notation.
Any zeros the decimal point moves past are
not significant, unless they are sandwiched
between two significant digits.
All other figures are taken as significant.
Counting Significant Digits
Any zeros that vanish when you convert a measurement to scientific
notation were not really significant figures. Consider the following
examples.
Converted to
Scientific
Significant
Measurement Notation
Figures
0.01234 kg
1.234 x 10-2 kg
4
Leading zeros (0.01234 kg)
just locate the decimal point.
They're never significant.
0.012340 kg
1.2340 x 10-2 kg
5
Notice that you didn't have
to move the decimal point
past the trailing zero
(0.012340 kg) so it doesn't
vanish and so is considered
significant.
0.000011010 m 1.1010 x 10-5 m
5
Again, the leading zeros
vanish but the trailing zero
doesn't.
4
Ditto.
0.3100 m
3.100 x 10-1 m
Counting Significant Digits
Measure
ment
Converted to
Scientific
Notation
321,010,000 3.2101 x 106
miles
miles
84,000 mg
8.4 x 104 mg
Signific
ant
Figures
5 (at Ignore commas. Here, the
least) decimal point is moved past the
trailing zeros (321,010,000 miles)
in the conversion to scientific
notation. They vanish and
should not be counted as
significant. The first zero
(321,010,000 miles) is significant,
though, because it's wedged
between two significant digits.
2 (at The decimal point moves past
least) the zeros (84,000 mg) in the
conversion. They should not be
counted as significant.
Counting Significant Digits
Measure
ment
Converted to
Scientific
Notation
32.00 mL 3.200 x 101 mL
302.120
lbs
3.02120 x 102
lbs
Signific
ant
Figures
4
The decimal point didn't move
past those last two zeros. They
are significant.
6
The decimal point didn't move
past the last zero (302.120 lbs, so
it is significant. The decimal point
did move past the 0 between the
two and the three, but it's wedged
between two significant digits, so
it's significant as well. All of the
figures in this measurment are
significant.
QUESTIONS
Fill in the blanks in the following table.
Number of
Measurement Significant Figures
a) 1. 0.010010 g ………………….
b) 10.00 g
………………….
c) 1010010 g
…………………..
ANSWERS
a) 0.010010 g has 5 significant figures. The trailing zero is
significant; the leading zeros are not.
b) 10.00 g has 4 significant figures. The trailing zeros are
significant. The decimal point does not move past them when the
number is converted to scientific notation.
c) 1010010 g has at least 6 significant figures. The trailing zero(s)
can not be counted as significant.
When are Zeros Significant?
From the previous frame, you know that whether a zero is significant
or not depends on just where it appears. Any zero that serves merely
to locate the decimal point is not significant. All of the possibilities
are covered by the following rules:
Rule
Examples
(Significant figures are red)
1. Zeros sandwiched between two
significant digits are always
significant.
1.0001 km
2501 kg
140.009 Mg
2. Trailing zeros to the right of the
3.0 m
decimal point are always significant. 12.000 µm
1000.0 µm (trailing zero is
significant by rule 2; others by
rule 1.)
When are Zeros Significant?
3. Leading zeros are never
significant.
0.0003 m
0.123 µm
0.0010100 µm (trailing zeros are
significant by rule 2;
sandwiched zeros by rule 1.)
4. Trailing zeros that all appear to the 3000 m
LEFT of the decimal point can not be 1230 µm
assumed to be significant.
92,900,000 miles
QUESTIONS
Fill in the blanks in the following table.
Minimum
Minimum
Number of
Number of
Measurement Significant Figures Measurement Significant Figures
a)
1010.010 g .......................
d) 32010.0 g ........................
b) 0.00302040 g .......................
e) 0.01030 g ........................
c)
f)
101000 g .......................
100 g ........................
ANSWERS
a) 1010.010 g has 7 significant figures. The trailing zero is right of
the decimal point and is significant; the zeros sandwiched between
the ones are also significant.
b) 0.00302040 g has 6 significant figures. The trailing zero are
significant, since it is to the right of the decimal point. The leading
zeros are not significant. The zeros that are sandwiched between
nonzero digits are significant.
c) 101000 g has at least 3 significant figures. The trailing zero(s) can
not be counted as significant.
d) 32010.0 g has 6 significant figures. The trailing zero is significant
because it is to the right of the decimal point; all of the other zeros
are sandwiched between two significant figures, so they're
significant, too.
e) 0.01030 g has 4 significant figures. The trailing zero(s) are
significant because they are to the right of the decimal point. Zero(s)
that are to the left of the first nonzero digit are NOT significant.
f) 100 g has at least 1 significant figures. The trailing zero(s) can not
be counted as significant.
Rounding Off
Often a recorded measurement that contains more than one uncertain digit
must be rounded off to the correct number of significant digits. For example,
if the last 3 figures in 1.5642 g are uncertain, the measurement should be
written as 1.56 g, so that only ONE uncertain digit is displayed.
Rules for Rounding Off Measurements
1. All digits to the right of the first uncertain digit have to be eliminated.
Look at the first digit that must be eliminated.
2. If the digit is greater than or equal to 5, round up.
1.35343 g rounded to 2 figures is 1.4 g.
1090 g rounded to 2 figures is 1.1 x 103 g.
2.34954 g rounded to 3 figures is 2.35 g.
3.
If the digit is less than 5, round down.
1.35343 g rounded to 4 figures is 1.353 g.
1090 g rounded to 1 figures is 1 x 103 g.
2.34954 g rounded to 5 figures is 2.3495 g.
Try these
a)
b)
c)
2.43479 rounded to 3 figures is
...........
b)1.756243 rounded to 4 figures is ...........
9.973451 rounded to 4 figures is
...........
Answers
a) 2.43479 should be rounded to 2.43. The fourth
figure is a 4, so the number is rounded down.
b) 1.756243 should be rounded to 1.756. The fifth
figure is a 2, so the number is rounded down.
c) 9.973451 should be rounded to 9.973. The fifth
figure is a 4, so the number is rounded down.
Counting Significant Digits for a Series of
Measurements
• Suppose you weigh a penny several times
and obtain the following masses.
• 2.5019 g
2.5023 g
2.5030 g
2.5037 g
2.5043 g
2.5009 g
• How many significant figures should the
average mass be reported to?
Counting Significant Digits for a Series of
Measurements
• The average can't be more precise than any of the
individual measurements. The exact average is
2.5026833333333333333333... g. Clearly this is not an
appropriate way to report the average, since it implies
more precision than any of the masses being averaged
actually have. Masses taken from the balance could be
estimated to the nearest tenth of a milligram (0.0001 g)
so the average can't be more precise than 2.5027 g.
• The last figure is the one and only uncertain figure.
Remember the definition of significant figures: all digits
up to and including the first uncertain digit. Look at the
penny weights above. Which digits are uncertain? The
last two places change from measurement to
measurement. If we want to write an average so that the
last figure recorded is the first (and only) uncertain
figure, it's best to write the average as 2.503 g.
Examples of Uncertainty calculations
1) Combining uncertainties in several
quantities: adding or subtracting
When one adds or subtracts several measurements together, one
simply adds together the uncertainties to find the uncertainty in the
sum.
Ex:
Dick and Jane are acrobats. Dick is 186 +/- 2 cm tall, and Jane is
147 +/- 3 cm tall. If Jane stands on top of Dick's head, how far is
her head above the ground?
Answer:
combined height = 186 cm + 147 cm = 333 cm uncertainty in
combined height = 2 cm + 3 cm = 5 cm combined height = 333 cm
+/- 5 cm
Now, if all the quantities have roughly the same magnitude and uncertainty -as in the example above -- the result makes perfect sense. But if one tries to
add together very different quantities, one ends up with a funny-looking
uncertainty.
EX:
Dick balances on his head a flea (ick!) instead of Jane. Using a pair of
calipers, Dick measures the flea to have a height of 0.020 cm +/- 0.003 cm.
ANSWER:
combined height = 186 cm + 0.020 cm = 186.020 cm uncertainty in combined
height = 2 cm + 0.003 cm = 2.003 cm
??? combined height = 186.020 cm +/- 2.003 cm ???
This doesn't make any sense! If we can't tell exactly where the top of Dick's
head is to within a couple of cm, what difference does it make if the flea is
0.020 cm or 0.021 cm tall?
In technical terms, the number of significant figures required to express
the sum of the two heights is far more than either measurement
justifies.
In plain English, the uncertainty in Dick's height swamps the uncertainty in
the flea's height; in fact, it swamps the flea's own height completely. A good
scientist would say
combined height = 186 cm +/- 2 cm
Combining uncertainties in several quantities:
multiplying and dividing
When one multiplies or divides several measurements together, one can often
determine the fractional (or percentage) uncertainty in the final result simply by
adding the uncertainties in the several quantities.
EXAMPLE:
Jane needs to calculate the volume of her pool, so that she knows how much water
she'll need to fill it. She measures the length, width, and height:
ANSWER:
length L = 5.56 +/- 0.14 meters = 5.56 m +/- 2.5%
width W = 3.12 +/- 0.08 meters = 3.12 m +/- 2.6%
depth D = 2.94 +/- 0.11 meters = 2.94 m +/- 3.7%
To calculate the volume, she multiplies together the length, width and depth:
volume = L * W * D = (5.56 m) * (3.12 m) * (2.94 m) = 51.00 m^3
In this situation, since each measurement enters the calculation as a multiple to
the first power (not squared or cubed), one can find the percentage uncertainty
in the result by adding together the percentage uncertainties in each
individual measurement:
percentage uncertainty in volume = (percentage uncertainty in L) + (percentage
uncertainty in W) + (percentage uncertainty in D) = 2.5% + 2.6% + 3.7% = 8.8%
Therefore, the uncertainty in the volume (expressed in cubic meters, rather than
a percentage) is
uncertainty in volume = (volume) * (percentage uncertainty in volume) = (51.00
m^3) * (8.8%) = 4.49 m^3
Therefore,
volume = 51.00 +/- 4.49 m^3 = 51.00 m +/- 8.8%