Course 3

5-3 Dimensional Analysis

Warm UpFind each unit rate.

1. Jump rope 192 times in 6 minutes

2. Four pounds of bananas for $2.36

3. 16 anchor bolts for $18.56

4. 288 movies on 9 shelves

32 jumps/min

$0.59/lb

$1.16/bolt

32 movies/shelf

Course 3

5-3 Dimensional Analysis

Learn to use one or more conversion factors to solve rate problems.

TB P. 224-228

Course 3

5-3 Dimensional Analysis

Vocabulary

conversion factor

Course 3

5-3 Dimensional Analysis

The process of converting from one unit to another is called dimensional analysis, or unit analysis. To convert units, multiply by one or more ratios of equal quantities called conversion factors.

For example, to convert inches to feet you would use the ratio below as a conversion factor.

1 ft12 in.

Course 3

5-3 Dimensional Analysis

Multiplying by a conversion factor is like multiplying by 1.

12 in.12 in.

1 ft1 ft

= , or = 11 ft12 in.

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5-3 Dimensional Analysis

Be sure to put the units you are converting to in the numerator and the units you are converting from in the denominator.

Caution!

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5-3 Dimensional Analysis

Find the appropriate factor for each conversion.

Additional Example 1: Finding Conversion Factors

A. feet to yards

B. pounds to ounces

1 yd3 ft

There are 3 feet in 1 yard. To convert feet to

yards, multiply the number of feet by .

16 oz1 lb

There are 16 ounces in 1 pound. To convert

pounds to ounces, multiply the number of

pounds by .

Course 3

5-3 Dimensional Analysis

The average American uses 580 pounds of paper per year. Find the number of pounds of paper the average American uses per month, to the nearest tenth.

Additional Example 2: Using Conversion Factors to Solve Problems

The problem gives the ratio 580 pounds to 1 year and asks for an answer in pounds per month.

580 lb 1 yr

1 yr 12 mo

580 lb 12 mo

=

= 48.3 lb per month

Multiply the ratio by the conversion factor.

Divide 580 by 12.

The average American uses 48.3 pounds of paper per month.

Cancel yr units. •yrmo= lb

molbyr

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5-3 Dimensional Analysis

Additional Example 3: Problem Solving Application

A car traveled 60 miles on a road in 2 hours. How many feet per second was the car traveling?

Course 3

5-3 Dimensional Analysis

11 Understand the Problem

The problem is stated in units of miles and hours. The question asks for the answer in units of feet and seconds. You will need to use several conversion factors.List the important information:

• Feet to miles5280 ft

1 mi

• Hours to minutes

• Minutes to seconds 1 min60 s

1 h60 min

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5-3 Dimensional Analysis

Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

22 Make a Plan

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5-3 Dimensional Analysis

First, convert 60 miles in 2 hours into a unit rate.

Solve33

60 mi2 h

= (60÷2) mi(2÷2) h

= 30 mi1 h

Create a single conversion factor to convert hours directly to seconds:

hours to seconds = • 1 min60 s

Set up the conversion factors.

minutes to seconds 1 min60 s

hours to minutes 1 h60 min

1 h60 min

1 h3600 s

=

30 mi1 h

• 5280 ft1 mi

• 1 h 3600 s

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5-3 Dimensional Analysis

Solve Continued33

Do not include the numbers yet.

Notice what happens to the units.

30 • 5280 ft • 1 1 • 1 • 3600 s

= 158,400 ft3600 s

= 44 ft1 s

The car was traveling 44 feet per second.

Simplify. Only remains.fts

Multiply.

mih

•ftmi

• hs

• •30 mi1 h

5280 ft1 mi

1 h 3600 s

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5-3 Dimensional Analysis

A rate of 44 ft/s is less than 50 ft/s. A rate of 60 miles in 2 hours is 30 mi/h or 0.5 mi/min.

44 Look Back

Since 0.5 mi/min is less than 3000 ft/60 s or 50 ft/s and 44 ft/s is less than 50 ft/s, then 44 ft/s is a reasonable answer.

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5-3 Dimensional Analysis

Additional Example 4: Physical Science Application

A strobe lamp can be used to measure the speed of an object. The lamp flashes every of a second. A camera records the object moving 52 cm between flashes. How fast is the object moving in m/s?

1 100

distance .time

Use rate =52 cm

1100

s

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5-3 Dimensional Analysis

It may help to eliminate the fraction first.

Additional Example 4 Continued

1 100

Multiply top and bottom by 100.

5200 cm1 s

=

52 cm1

100s

= 100 • 52 cm 1

100 s 100 •

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5-3 Dimensional Analysis

Now convert centimeters to meters.

Additional Example 4 Continued

5200 cm1 s

Multiply by the conversion factor.

5200 m100 s

=52 m1 s

=

The object is traveling 52 m/s.

5200 cm1 s

= • 1 m100 cm

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5-3 Dimensional Analysis

Lesson Quiz

Find the appropriate factor for each conversion.

1. kilograms to grams

2. pints to gallons

3. You drive 136 miles from your house to your aunt’s

house at the lake. You use 8 gallons of gas. How

many yards does your car get to the gallon?

4. A cheetah was timed running 200 yards in 6 seconds.

What was the average speed in miles per hour?

1000 gkg

1 gal8 pt

29,920 ydgal

≈ 68 mi/h