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

Warm Up Find 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

Problem of the Day Replace each • with a digit from 0 to 6 to make equivalent ratios. Use each digit only once. Possible answer: 13 65 4 20 = •• • •• =

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

Vocabulary conversion factor

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 ft 12 in.

Multiplying by a conversion factor is like multiplying by 1. 1 ft 12 in. 12 in. 1 ft = , or = 1

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

Additional Example 1: Finding Conversion Factors Find the appropriate factor for each conversion. A. feet to yards B. pounds to ounces 1 yd 3 ft There are 3 feet in 1 yard. To convert feet to yards, multiply the number of feet by . 16 oz 1 lb There are 16 ounces in 1 pound. To convert pounds to ounces, multiply the number of pounds by .

Check It Out: Example 1 Find the appropriate factor for each conversion. A. minutes to seconds B. hours to days 60 sec 1 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 .

Additional Example 2: Using Conversion Factors to Solve Problems 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. 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 Multiply the ratio by the conversion factor Cancel yr units. • yr mo = lb 580 lb 12 mo = = 48.3 lb per month Divide 580 by 12. The average American uses 48.3 pounds of paper per month.

Multiply the ratio by the conversion factor Check It Out: Example 2 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 Multiply the ratio by the conversion factor Cancel yr units. • yr mo = mi 23,000 mi 12 mo = = 1916.6 per month Divide 23,000 by 12. Sam drives his car about 1917 miles per month.

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?

Understand the Problem 1 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: 5280 ft 1 mi • Feet to miles 1 h 60 min • Hours to minutes 1 min 60 s • Minutes to seconds

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

First, convert 60 miles in 2 hours into a unit rate. Solve 3 First, convert 60 miles in 2 hours into a unit rate. 60 mi 2 h (60÷2) mi (2÷2) h = 30 mi 1 h = Create a single conversion factor to convert hours directly to seconds: minutes to seconds 1 min 60 s hours to minutes 1 h 60 min 1 h 60 min 1 min 60 s 1 h 3600 s = hours to seconds = • 30 mi 1 h • 5280 ft 1 mi 3600 s Set up the conversion factors.

Do not include the numbers yet. 3 Solve Continued Do not include the numbers yet. Notice what happens to the units. mi h • ft s Simplify. Only remains. ft s • 30 mi 1 h 5280 ft 1 mi 3600 s Multiply. 30 • 5280 ft • 1 1 • 1 • 3600 s = 158,400 ft 3600 s = 44 ft 1 s The car was traveling 44 feet per second.

Since 0.5 mi/min is less than 3000 ft/ 4 Look Back 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. 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.

Check It Out: Example 3 A train traveled 180 miles on a railroad track in 4 hours. How many feet per second was the train traveling?

Understand the Problem 1 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: 5280 ft 1 mi • Feet to miles 1 h 60 min • Hours to minutes 1 min 60 s • Minutes to seconds

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

First, convert 180 miles in 4 hours into a unit rate. Solve 3 First, convert 180 miles in 4 hours into a unit rate. 180 mi 4 h (180 ÷ 4) mi (4 ÷ 4) h 45 mi 1 h = = Create a single conversion factor to convert hours directly to seconds: minutes to seconds 1 min 60 s hours to minutes 1 h 60 min 1 h 60 min 1 min 60 s 1 h 3600 s = hours to seconds = • 45 mi 1 h • 5280 ft 1 mi 3600 s Set up the conversion factors.

Do not include the numbers yet. 3 Solve Continued Do not include the numbers yet. Notice what happens to the units. mi h • ft s Simplify. Only remains. ft s • 45 mi 1 h 5280 ft 1 mi 3600 s Multiply. 45 • 5280 ft • 1 1 • 1 • 3600 s = 237,600 ft 3600 s = 66 ft 1 s The train was traveling 66 feet per second.

4 Look Back 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. 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.

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 52 cm 1 100 s distance . time Use rate =

Additional Example 4 Continued 1 100 It may help to eliminate the fraction first. 52 cm 1 100 s = 100 • 52 cm 1 100 s 100 • Multiply top and bottom by 100. 5200 cm 1 s =

Additional Example 4 Continued Now convert centimeters to meters. 5200 cm 1 s 5200 cm 1 s = • 1 m 100 cm Multiply by the conversion factor. 5200 m 100 s = 52 m 1 s = The object is traveling 52 m/s.

Check It Out: Example 4 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 65 cm 1 100 s distance . time Use rate =

Check It Out: Example 4 Continued 1 100 It may help to eliminate the fraction first. 65 cm 1 100 s = 100 • 65 cm 1 100 s 100 • Multiply top and bottom by 100. 6500 cm 1 s =

Check It Out: Example 4 Continued Now convert centimeters to meters. 6500 cm 1 s 6500 cm 1 s = • 1 m 100 cm Multiply by the conversion factor. 6500 m 100 s = 65 m 1 s = The object is traveling 65 m/s.

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 g kg 1 gal 8 pt 29,920 yd gal ≈ 68 mi/h