1. Course 3 5-3 Dimensional Analysis 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

2. Course 3 5-3 Dimensional Analysis Learn to use one or more conversion factors to solve rate problems. TB P. 224-228

3. Course 3 5-3 Dimensional Analysis Vocabulary conversion factor

4. 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 ft 12 in.

5. Course 3 5-3 Dimensional Analysis Multiplying by a conversion factor is like multiplying by 1. 12 in. 1 ft =, or=1 1 ft 12 in.

6. Course 3 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!

7. Course 3 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 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.

8. 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. yr mo = lb mo lb yr

9. Course 3 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?

10. Course 3 5-3 Dimensional Analysis 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: Feet to miles 5280 ft 1 mi Hours to minutes Minutes to seconds 1 min 60 s 1 h 60 min

11. Course 3 5-3 Dimensional Analysis Multiply by each conversion factor separately, or simplify the problem and multiply by several conversion factors at once. 2 Make a Plan

12. Course 3 5-3 Dimensional Analysis First, convert 60 miles in 2 hours into a unit rate. Solve 3 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: hours to seconds = 1 min 60 s Set up the conversion factors. minutes to seconds 1 min 60 s hours to minutes 1 h 60 min 1 h 60 min 1 h 3600 s = 30 mi 1 h 5280 ft 1 mi 1 h 3600 s

13. Course 3 5-3 Dimensional Analysis Solve Continued 3 Do not include the numbers yet. Notice what happens to the units. 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. Simplify. Only remains. ft s Multiply. mi h ft mi hshs 30 mi 1 h 5280 ft 1 mi 1 h 3600 s

14. Course 3 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. 4 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.

15. Course 3 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 1 100 s

16. Course 3 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 cm 1 s = 52 cm 1 100 s = 100 52 cm 1 100 s

17. Course 3 5-3 Dimensional Analysis Now convert centimeters to meters. Additional Example 4 Continued 5200 cm 1 s Multiply by the conversion factor. 5200 m 100 s = 52 m 1 s = The object is traveling 52 m/s. 5200 cm 1 s = 1 m 100 cm

18. Course 3 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 g kg 1 gal 8 pt 29,920 yd gal ≈ 68 mi/h