5-4 Dimensional Analysis

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California StandardsCalifornia Standards

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5-4 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

5-4 Dimensional Analysis

MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems (e.g., miles per hour and feet per second, cubic inches to cubic centimeters).

California Standards

5-4 Dimensional Analysis

Vocabulary

conversion factor

5-4 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 conversion factors. A conversion factor is a ratio of two quantities that are equal but use different units.

For example, to convert inches to feet

you would use the ratio as a conversion factor.

1 ft12 in.

5-4 Dimensional Analysis

Multiplying by a conversion factor is like multiplying by 1.

12 in.12 in.

1 ft1 ft

= = = 11 ft12 in.

5-4 Dimensional Analysis

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

Additional Example 1: Using Conversion Factors to Solve Problems

Convert the rate 580 pounds per year to pounds per month.

580 lb 1 yr

1 yr 12 mo

580 lb 12 mo

48.3 lb per month

To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity.

Divide 580 by 12.

The average American uses 48.3 pounds of paper per month.

Divide out like units. •yrmo=

lbmo

lb yr

5-4 Dimensional AnalysisCheck It Out! Example 1

Sam drives his car 23,040 miles per year. Find this rate in the number of miles driven per month, to the nearest mile.

Convert the rate 23,040 miles per year to miles per month.

23,040 mi 1 yr

1 yr 12 mo

23,040 mi 12 mo

=

= 1920 per month Divide 23,040 by 12.

Sam drives his car 1920 miles per month.

Divide out units. •yrmo

= mimo

miyr

To convert the second amount in a rate, multiply by the conversion factor with that unit in the first quantity.

5-4 Dimensional AnalysisAdditional Example 2: Problem Solving Application

A car traveled 60 miles on a road in 2 hours. Find this rate in feet per second.

5-4 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

5-4 Dimensional Analysis

You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

22 Make a Plan

5-4 Dimensional Analysis

Convert to miles per hour.

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

5-4 Dimensional Analysis

Solve Continued33

30 • 5280 ft • 1 1 • 1 • 3600 s

= 158,400 ft3600 s

= 44 ft1 s

The car was traveling 44 feet per second.

Divide out like units.• •30 mi1 h

5280 ft1 mi

1 h 3600 s=

=

5-4 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.

5-4 Dimensional AnalysisCheck It Out! Example 2

A train traveled 180 miles on a railroad track in 4 hours. Find this rate in feet per second.

5-4 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

5-4 Dimensional Analysis

22 Make a Plan

You know the conversion factor that converts miles to feet. So multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once.

5-4 Dimensional Analysis

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

5-4 Dimensional Analysis

Solve Continued33

45 • 5280 ft • 1 1 • 1 • 3600 s

= 237,600 ft3600 s

= 66 ft1 s

The train was traveling 66 feet per second.

Divide out like units.• •45 mi1 h

5280 ft1 mi

1 h 3600 s=

=

5-4 Dimensional Analysis

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.

5-4 Dimensional AnalysisAdditional Example 3: 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? Use dimensional analysis to check the reasonableness of your answer.

1 100

distance .time

Use rate =52 cm

1100

s

5-4 Dimensional Analysis

It may help to eliminate the fraction first.

Additional Example 3 Continued

1 100

Multiply numerator and denominator by 100.

5200 cm1 s

=

52 cm1

100s

= 100 • 52 cm 1

100 s 100 •

5-4 Dimensional Analysis

Convert centimeters to meters to see if the answer is reasonable.

Additional Example 3 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

5-4 Dimensional AnalysisCheck It Out! Example 3

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

5-4 Dimensional Analysis

It may help to eliminate the fraction first.

Check It Out! Example 3 Continued

1 100

Multiply numerator and denominator by 100.

6500 cm1 s

=

65 cm1

100s

= 100 • 65 cm 1

100 s 100 •

5-4 Dimensional Analysis

Convert centimeters to meters to see if the answer is reasonable.

Check It Out! Example 3 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

5-4 Dimensional Analysis

Lesson Quiz

1. 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?

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

What was its average speed in miles per hour?

29,920 ydgal

≈ 68 mi/h