10 gallons = $15. 5 gallons.

Transcription

1 V. Analyze units Unit analysis, also called dimensional analysis, is a problem solving technique that pays attention to the units of measure employed in a problem situation, using contextual information about the problem to point to a solution. In any situation involving measurement, units must be included to make sense of the answer. For example, suppose your boss asked you to find out how much fencing you would need to enclose the company parking lot. You did the research and responded by saying you needed 45 to do the job. So your boss bought 45 yards of fence material at a one-time only sale price. Unfortunately, you meant you needed only 45 feet of fencing. You can be sure the boss will not be pleased to have 290 feet of unused fencing material left over! A ratio is a comparison of two numbers. Ratios are often written as fractions, so when gasoline prices are listed in units of dollars per gallon, we often represent such prices in fractional form: $3/gal, say. This is equivalent to reporting that $30 is spent on 0 gallons, or $5 for lons. Notice that these quantities are also equivalent fractions: $3 lon $30 0 gallons $5 lons. Example: Toni drove les in 2 hours and used lons of gas. Calculate the six possible pairs of ratios of the three quantities in this problem. Solution: These ratios are Miles per hour Hours per mile Gallons per hour Hours per gallon Gallons per mile Miles per gallon 40 mi hr hr mi 2. hr 0.4 hr mi 6 mi 40 mi hr 40 mph; hr mi ; 2. hr ; 0.4 hr gal ; mi ; mi 6 6 mpg. gal 6

2 The second of the measurements in the solution above, hr per mi, may not mean anything to you since we don t usually report such a short period of time in hours. But if we convert this quantity into seconds, this would clarify its meaning. We do this by means of a unit conversion, performed by multiplying the quantity we wish to convert by a quantity equal to, judiciously chosen to represent a useful ratio or product of ratios for the purpose of cancelling like units. Here, a useful product of ratios would be 60 min 60 sec hr min, because multiplying hr per mi by this quantity allows us to cancel units from the numerators and denominators of the fractions, leaving behind only the unit we desire: hr 60 min 60 sec mi hr min 90 sec mi. The quantity 90 sec per mi makes more sense than hr per mi (since it tells us that Toni covered the distance of a mile in 90 sec. Example: A leaky faucet drips fluid ounce of water every 30 seconds. How many gallons of water will leak from this faucet in a 30-day month? Solution: To convert the drip rate from ounces per second into gallons per month, we select a sequence of conversion ratios that swap the given units into the desired units, then multiply to get the answer: oz 30 sec 60 sec 60 min min hr 24 hr day 30 day mo cup 8 oz qt 4 cup 67 4 qt mo. Example: In Major League Baseball, some pitchers throw 00 mph fastballs. How much time does a batter have to react to such a pitch? Use the fact that the pitcher s mound is 60 feet, 6 inches from home plate. Solution: The given information involves units of miles, hours, feet and inches, but the answer will surely require a small unit of time, like seconds. Given the speed of the ball and the distance it must travel from pitcher to batter, and recognizing that speed is in units of distance per time, we will determine the time it takes the ball to travel this distance by dividing the 7

3 distance it travels by its speed, or equivalently, by multiplying the distance it travels by the reciprocal of its speed: 60.5 ft mi hr 5280 ft 00 mi 60 min hr 60 sec sec. min Some problems require the use of compound units. For instance, the unit passenger-mile shows up in transportation studies, both as a measure of the efficiency of public transportation and in traffic safety statistics. You might see statistics on the number of deaths per billion passenger-miles or accidents per billion passenger-miles. Just what is a passenger-mile? It represents one passenger traveling one mile. For example, a car with one passenger traveling 2 miles racks up 2 passenger-miles. A car with 3 passengers traveling 5 miles accounts for 5 passenger-miles. An airplane that travels 2500 miles with 400 passengers aboard accounts for million passenger-miles. Example: Gerry and three friends regularly drive a 208-mile trip between their home town and the university they attend. Gerry s car gets 35 miles per gallon and gas currently costs about $3.5 per gallon. If they drive this trip at an average speed of 50 mph, then find each of the following quantities that characterize this travel situation:. The amount of gasoline they use; 2. The time of travel; 3. Their average speed in feet per second; 4. The cost of travel in cents per passenger-mile. Solution: The amount of gasoline will best be measured in gallons. We know the fuel efficiency of the car (in mi/gal) and the distance of travel (in mi). To get a quantity in units of gallons requires dividing miles by miles per gallon: 35 mi 35 mi 5.94 gal. To find time of travel, we can combine speed (in mph) with distance (in mi). The resulting time will be in hours if we divide distance by speed: 8

4 50 mi hr hr 50 mi 0.6 hr 4 hr hr 4 hr hr 60 min hr 4 hr +0 min. To convert speed from mph to ft/sec, we multiply by forms of that introduce the conversion factors from miles to feet and hours to seconds: 50 mi hr 5280 ft mi hr 60 min min 60 sec ft 3600 sec 73.3 ft/sec. To find cents per passenger-mile, we need the cost of the trip in cents and the number of passenger-miles. For cost, we know that gasoline costs $3.5 per gallon, their car gets 35 miles per gallon, and they travel les. These measures can be combined to eliminate every unit but dollars, which can easily be converted to cents: 3.5 dollars 35 mi 00 cents dollar 872 cents. Then, since 4 people ride les, there are passenger-miles, whence the cost per passenger-mile is 872 cents 832 passenger - miles 2.25 cents per passenger- mile. Example: Janice, Shane, Rose, and Gina are going to be paid $84.70 for cleaning up Mr. Rogers neighborhood. They each worked 5 hours, except Rose, who was 45 minutes late. How much should each be paid? Solution: Rose contributed 4.5 hrs of labor, while the other three contributed 5 hrs each (or 5 hrs for all three), for a total of 9.5 hrs. Thus $ hrs $4.40 per hr. 9

5 To find the fair share for each worker, multiply his or her time at work by the $4.40 per hour rate. So for Janice, Gina, and Shane, this is $4.40 hr 5 hr $22.00, whereas for Rose, the fair share is only $4.40 hr 4.5 hr $