MTH 065 Class notes Lecture 9 (2.6)
Section 2.6: Using Dimensional Analysis to solve applications
Let’s start by reviewing the technique Dimensional Analysis
Review of Unit Fractions
The method of dimensional analysis uses fractions to perform conversions like the ones
described above. It relies upon two properties of multiplication of fractions.
1. To multiply two fractions, we multiply straight across, numerator times numerator and
denominator times denominator. (See Appendix A for a review of fractions.)
2. If we multiply a fraction by 1 or any fraction equivalent to 1, the result is equivalent to the
original fraction. For example,
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In the technique of dimensional analysis, these two ideas are combined to perform
conversions. We use fractions called unit fractions.
Some examples of unit fractions are:
Note: Appendix F has conversion tables to help you with this section.
Dimensional Analysis
The key to dimensional analysis is choosing the appropriate unit fraction.
Always multiply by a unit fraction that allows us to "cancel" the unit we want to eliminate and
leaves a unit that we want.
Practice 1
Convert 4.85 meters to centimeters.
When you use several unit fractions in dimensional analysis, the process is called “chaining”.
Practice 2
How many gallons are in a container that holds 16.4 liters?
Dimensional Analysis with Area
To convert from one unit of area to another unit of area, we usually use the conversion factor two
times.
For example: The exponent of 2 on the unit is a reminder that there are two dimensions to
consider.
Practice 3:
Use dimensional analysis to solve each of the following problems:
(a) How many square inches are in 10 square feet?
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(b) An underground tank holds 250 gallons. Joe wants to compare the cost of filling it with
gravel to the cost digging it up. How many yards of gravel will it take to fill the tank?
Round your answer to one decimal place. (Note: in construction, "yards" of gravel, sand,
concrete, etc., is understood to mean "cubic yards".)
Let’s start by translating this problem:
From Appendix F, the conversion for gallons to cubic space is:
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Dimensional Analysis with Rates of Change
To convert from one mixed unit in fraction form to a different mixed unit in fraction form, you
may eliminate the units either in the numerator or in the denominator first. Just multiply by as
many unit fractions as are necessary to convert both.
Practice 4
Use dimensional analysis to solve each of the following problems:
(a) Lynn walks for exercise every day. She averages 6 km/hr. What is her speed in m/sec?
(b) The air pressure in the tires of a particular van is 35 psi (pounds per square inch). What is
the pressure in kilograms per square meter (kg/m2).
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Practice 5
Use dimensional analysis to solve the following problem:
The price of gasoline in Canada is 90 Canadian cents per liter. If one US dollar is equivalent to
$1.70 in Canadian money, how much does a gallon of gasoline cost in US dollars?
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