5-4 Dimensional Analysis

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpUse cross products to solve the proportions.

Course 2

5-4 Dimensional Analysis

1.

2.

3.

4.

n = 25

r = 24

k = 36

x = 1240x

=309

38

= r64

917

= k68

610

= 15n

Problem of the Day

The sum of four consecutive integers is 182. What are the four numbers?

44, 45, 46, and 47

Course 2

5-4 Dimensional Analysis

Learn to use dimensional analysis to make unit conversions.

Course 2

5-4 Dimensional Analysis

Vocabulary

unit conversion factor

Insert Lesson Title Here

Course 2

5-4 Dimensional Analysis

Course 2

5-4 Dimensional Analysis

You can use a unit conversion factor to change, or convert, measurements from one unit to another. A unit conversion factor is a fraction in which the numerator and denominator represent the same quantity, but in different units. The fraction below is a unit conversion factor that can be used to convert miles to feet. Notice that it can be simplified to one.

5,280 ft1 mi

= 5,280 ft5,280 ft

= 1

Course 2

5-4 Dimensional Analysis

Multiplying a quantity by a unit conversion factor changes only its units, not its value. The process of choosing an appropriate conversion factor is called dimensional analysis.

Course 2

5-4 Dimensional Analysis

When choosing a unit conversion factor, choose the one that cancels the units you want to change and replaces them with the units you want.

Helpful Hint

An oil drum holds 55 gallons. How many quarts of oil will fill the drum? Use a unit conversion factor to convert the units.

Additional Example 1: Making Unit Conversions

Course 2

5-4 Dimensional Analysis

55 gal ·

= 220 qt

220 quarts of oil will fill the drum.

Multiply.

One gallon equals 4 quarts so use the conversion factor or . Choose the second one so the1 gal

4 qt4 qt1 gal

gallon units will “cancel.”

=4 qt1 gal

55 · 4 qt1

Try This: Example 1

Course 2

5-4 Dimensional Analysis

7 qt ·

= 14 pt

Multiply.

One quart equals 2 pints so use the conversion factor or . Choose the second one so the1 qt

2 pt2 pt1 qt

quarts units will “cancel.”

=2 pt1 qt

7 · 2 pt1

An ice cream recipe calls for 7 quarts of milk. How many pints of milk is this? Use a unit conversion factor to convert the units.

7 quarts of milk is 14 pints.

Use a unit conversion factor to convert the units within each rate.

Additional Example 2A: Making Rate Conversions

Course 2

5-4 Dimensional Analysis

A. If orange juice sells for $1.28 per gallon,what is the cost per ounce?

$1.28gal

· 1 gal4 qt

· 1 qt32 oz

= $1.28 · 1 · 11 · 4 · 32 oz

Multiply.

= $1.28128 oz

= $1.28 ÷ 128

128 oz ÷ 128

=$0.011 oz

$1.28 per gallon is $0.01 per ounce.

Insert Lesson Title Here

Use a unit conversion factor to convert the units within each rate.

B. Convert 80 miles per hour to miles per minute.

80 mi1 hr

· 80 mi · 11 · 60 min

Multiply.

= 80 mi ÷ 6060 min ÷ 60

≈ 1.33 mi 1 min

80 miles per hour is about 1.33 miles per minute.

Additional Example 2B: Making Rate Conversions

Course 2

5-4 Dimensional Analysis

1 hr60 min

=

Use a unit conversion factor to convert the units within each rate.

Course 2

5-4 Dimensional Analysis

$2.24gal

· 1 gal4 qt

· 1 qt2 pt

= $2.24 · 1 · 11 · 4 · 2 pt

Multiply.

= $2.24 8 pt

=$2.248 pt

= $0.28 1 pt

$2.24 per gallon is $0.28 per pint.

Try This: Example 2A

If milk sells for $2.24 per gallon, what is the cost per pint?

Insert Lesson Title Here

Use a unit conversion factor to convert the units within each rate.

B. Convert 50 miles per hour to miles per minute.

50 mi1 hr

· 50 mi · 11 · 60 min

Multiply.

= 50 mi 60 min

≈ 0.833 mi

1 min

50 miles per hour is about 0.83 miles per minute.

Try This: Example 2B

Course 2

5-4 Dimensional Analysis

1 hr60 min

=

The Mare Orientale crater on the Moon is more than 620 miles across. How many meters is this?

Additional Example 3: Measurement Application

Course 2

5-4 Dimensional Analysis

Use unit conversion factors that convert miles to to kilometers, and then kilometers to meters. Onekilometer is equivalent to 0.62 mile.

620 mi · 620 · 1 · 1,000 m0.62

= 620,000 m ÷ 0.620.62 ÷ 0.62

= 1,000,000 m

620 miles is 1,000,000 meters.

·1 km

0.62 mi1,000 m

1 km=

Course 2

5-4 Dimensional Analysis

Use unit conversion factors that convert pounds to to kilograms, and then kilograms to grams. Onekilogram is equivalent to 2.2 pounds.

5 lb · 5 · 1 · 1,000 g2.2

= 5,000 g ÷ 2.22.2 ÷ 2.2

≈ 2,273 g

5 pounds is about 2,273 grams.

·1 kg

2.2 lb1,000 g

1 kg=

Try This: Example 3

Mary went to the grocery store to buy 5 pounds of peaches. How many grams is this?

Lesson Quiz

Insert Lesson Title Here

Course 2

5-4 Dimensional Analysis

Use a unit conversion factor to convert the units.

1. Football fields are 100 yards long. How many feet is that?

2. In biology lab you measure a grasshopper’s wing span to be 3 inches long. How many centimeters is this?

Use a unit conversion factor to convert the unitswithin a rate.

3. On a freeway a car’s speed is 62 miles per hour.What speed is that in feet per hour?

4. If you are paid $7.50 per hour to watch your neighbor’s children, how much are you paid perminute?

300 feet

7.62 cm

327,360 ft/h

12.5 cents per minute