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Calculations 1

Measurements
always have some limitation on their precision. When you do calculations with
measurements (which you will undoubtedly have to do) the results will need to
reflect the limitations of the measurments upon which they are based. The
results will have their precision

limited by the
precision of the least precise measurement. The measurement with the smallest
number of significant figures is the least precise measurement. A result
calculated based on that measurement cannot have higher precision than the
measurement you started with. Put another way: a chain is only as strong as
the weakest link. A relay race team is only as fast as its slowest member.

### Measurements and Calculations

Make the following measurements and record the information below. Performs the
indicated calculations and prepare to share the data with the rest of the
class.

- With your group measure the area of the top of the
writing desks where you sit during normal classes. Measure using a meter stick
and report the length and width in cm and the area in cm
^{2}. - Now measure the area of a lab bench. Do not take the
time to subtract the area of the sink, just get measurements of the length and
width (again, in cm) and the area (cm
^{2}). - Measure the area of this piece of paper. Again, stick to cm units.
- Find the volume of an office supply box provided by
your instructor. Report the result in cm
^{3} - Find the volume of another box. Again, use cm
^{3}. (By the way, what other unit of volume could be used to report the result?)

Desk data:

L = W = and A =

Bench data:

L = W = and A =

Paper data:

L = W = and A =

Box 1 data: Identity of box:

L = W = H =

and V =

Box 2 data: Identity of box:

L = W = H =

and V =

If a measurement of length is off by 0.01 cm (the maximum precision of a
typical ruler) then a calculation of area would be off by 0.01 cm^{2}.
If you measure an area 2.25 cm on one side and 3.25 cm on the other then the
area is 7.31 cm^{2}. The calculator says that the area is 7.3125
cm^{2}. This is wrong because it *means* that you know the area
to a precision of ±0.0001 cm^{2}. Every decimal place added to
a result implies an improvement in precision of a factor of ten. So a
measurement precise to ±0.0001 is 100 times more precise than a
measurement precise to ±0.01. You can’t possibly have a precision
that is 100 times better after doing the calculation than you had in the
original measurements.

Basically, the important thing to know is how much to round off the results
of calculations. Here are the rules:

- The
*least*number of significant figures in any number in your calculation determines the number of significant figures in the answer. - Pure numbers (such as exact conversion factors and defined quantities) do not affect the number of significant figures.

Examples:

Calculator gives: 8.55

Correct answer is: 8.6

8.6 is the correct answer because of the rule governing significant figures in calculations. 2.5 has two significant figures and 3.42 has three. Since 2.5 has the least number of significant figures it is the number that determines how many significant figures are in the answer.

Calculator gives: 17.9459

Correct answer is: 17.9

You may have thought the answer should be 18. but this is wrong because 3.10 has three significant figures, not two.

Calculator gives: 12.00379

Correct answer is: 12.0

The number of significant figures in the answer must be three because 2.1 has only three significant figures. The trailing zero in 12.0 is important because it is significant. Remember the rules for significant zeros: Trailing zeros in the decimal portion only are significant.

Examples of pure numbers/exact figures: π (use the π button of your calculator), 12 in. = 1 ft, constants in formulas, 100 years = 1 century, 100 cm = 1 m, numbers of things (how many crayons in a box).

One more note: in this activity please remember not only to pay attention to the correct number of significant figures in the answer. Also, make sure to show units at each step of each calculation!

- 2.33 cm × 6.085 cm × 2.1 cm

Calculator’s answer:

Correct answer:

No. of s.f.:

- 4/3 × π × 4.598 cm
^{3}

Calculator’s answer:

Correct answer:

No. of s.f.:

- 5.789 ÷ 2.1

Calculator’s answer:

Correct answer:

No. of s.f.: - 56,000 ÷ 81.

Calculator’s answer:

Correct answer:

No. of s.f.: - Convert 3.0 m to mm

- Convert 4.5 ft to m

- Convert 90 mi/hr to m/s

- Convert 9.876 km to mi

- Find the volume of a rectangular solid with the following measurements: L = 40.5 cm W = 49.2 cm and H = 51.34 cm.
- Convert the answer to the previous problem from
cm
^{3}to in^{3}.

- A tennis ball has a radius of 3.33 cm. Find the
volume in cm
^{3}(V_{sphere}= 4/3πr^{3}). - You are hoping to attract foreign buyers to buy
apartments that you own in New York City. Since nearly all countries in the
world use meters instead of feet you must list the floor space in the
apartments in both ft
^{2}and m^{2}. Your apartments have the following areas: 1,500 square feet, 1,620 square feet, and 1,789 square feet. What are their areas in square meters? - An important tool in chemical identification is the
measurement of density. Density is defined as mass ÷ volume. What is
the density of a substance, in g/cm
^{3}, that has a mass of 14.76 g and a volume of 1.65 cm^{3}? - You find that the mass of 50 sheets of paper is 227 g. What is the mass of a single piece of paper?
- The diameter of a circle is 2.1 m. What is its area?
(A
_{circle}= πr^{2}). - The dimensions of a small box are 14.2 cm x 12.5 cm x 2.76 cm. Sixty-four of these boxes fit into a packing crate. What is the volume of the packing crate?

- Our classroom is about 12 m long and 9 m wide. When there are 19 people in the room how many square meters does each person have to move around in?