Course 3 5-3 Dimensional Analysis 5-3 Dimensional Analysis Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

5-3 Dimensional Analysis5-3 Dimensional Analysis

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

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

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Problem of the Day

Replace each • with a digit from 0 to 6 to make equivalent ratios. Use each digit only once. ••••

•••

= Possible answer:1365

420

=

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

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Vocabulary

conversion factor

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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.

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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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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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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 .

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Check It Out: Example 1


A. minutes to seconds

B. hours to days

60 sec1 min

There are 60 seconds in 1 minute. To convert

minutes to seconds, multiply the number of

minutes by .

1 day 24 h

There are 24 hours in 1 day. To convert

hours to days, multiply the number hours by

.

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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

Course 3



Sam drives his car 23,000 miles per year. Find the number of miles he drives per month, to the nearest mile.

The problem gives the ratio 23,000 miles to 1 year and asks for an answer in miles per month.

23,000 mi 1 yr

1 yr 12 mo

23,000 mi 12 mo

=

= 1916.6 per month

Multiply the ratio by the conversion factor

Divide 23,000 by 12.

Sam drives his car about 1917 miles per month.

Cancel yr units. •yrmo

= mimo

miyr

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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?

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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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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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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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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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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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A train traveled 180 miles on a railroad track in 4 hours. How many feet per second was the train traveling?

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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

Course 3


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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First, convert 180 miles in 4 hours into a unit rate.

Solve33

180 mi4 h

= (180 ÷ 4) mi(4 ÷ 4) h

= 45 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

=

45 mi1 h

• 5280 ft1 mi

• 1 h 3600 s

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Solve Continued33

Do not include the numbers yet.

Notice what happens to the units.

45 • 5280 ft • 1 1 • 1 • 3600 s

= 237,600 ft3600 s

= 66 ft1 s

The train was traveling 66 feet per second.

Simplify. Only remains.fts

Multiply.

mih

•ftmi

• hs

• •45 mi1 h

5280 ft1 mi

1 h 3600 s

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A rate of 66 ft/s is more than 50 ft/s. A rate of 180 miles in 4 hours is 45 mi/h or 0.75 mi/min.

44 Look Back

Since 0.75 mi/min is more than 3000 ft/60 s or 50 ft/s and 66 ft/s is more than 50 ft/s, then 66 ft/s is a reasonable answer.

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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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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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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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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 65 cm between flashes. How fast is the object moving in m/s?

1 100

distance .time

Use rate =65 cm1

100s

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It may help to eliminate the fraction first.

Check It Out: Example 4 Continued

1 100

Multiply top and bottom by 100.

6500 cm1 s

=

65 cm1

100s

= 100 • 65 cm 1

100 s 100 •

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Now convert centimeters to meters.

Check It Out: Example 4 Continued

6500 cm1 s

Multiply by the conversion factor.

6500 m100 s

=65 m1 s

=

The object is traveling 65 m/s.

6500 cm1 s

= • 1 m100 cm

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Lesson Quiz


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

Documents

Course 3 5-3 Dimensional Analysis 5-3 Dimensional Analysis Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation