Handout Unit Conversions (Dimensional Analysis)


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1 Handout Unit Conversions (Dimensional Analysis) The Metric System had its beginnings back in 670 by a mathematician called Gabriel Mouton. The modern version, (since 960) is correctly called "International System of Units" or "SI" (from the French "Système International"). The metric system has been officially sanctioned for use in the United States since 866, but it remains the only industrialized country that has not adopted the metric system as its official system of measurement. Many sources also cite Liberia and Burma as the only other countries not to have done so. This section will cover conversions () selected units in the metric and American systems, () compound or derived measures, and () between metric and American systems. Also, (4) applications using conversions will be presented. The metric system is based on a set of basic units and prefixes representing powers of 0. Basic metric units length meter (m) mass gram (g) volume liter (L) time second (s) temperature Celsius ( C) or Kelvin ( K) where C=K7.5 Prefixes (The units that are in bold are the ones that are most commonly used.) prefix symbol value giga G,000,000,000 = 0 9 ( a billion) mega M,000,000 = 0 6 ( a million) kilo k,000 = 0 hecto h 00 = 0 deca da 0 deci centi milli micro nano d c m μ (the Greek letter mu) n = 0. = 00 = 0.0 = 000 = 0.00 = 0  (a thousandth),000 = = 06 (a millionth),000,000 = = 09 (a billionth),000,000,000
2 To get a sense of the size of the basic units of meter, gram and liter consider the following examples. The standard height of a door handle or knob is meter. The weight of a penny is.5 grams. Almost everyone is familiar with liter bottles of soda. The next set of examples illustrates the use of some of the most used prefixes. Examples:. milligram = gram = 0.00 gram, or mg =,000, mg=g. milliliter= liter=0.00 liter, or ml=,000,000 L=0.00 L 4.,000 ml = L g = 0.00 gram 5. centimeter= 00 meter=0.0 meter, or cm = 00 m = 0.0 m 6. kilogram =,000 grams, or kg =,000 g 7. kilometer =,000 meters, or km = 000 m 8. kilobyte = 000 bytes (a byte is a unit used in computers to express the length of a word.) 9. megabyte =,000,000 bytes = 000 kilobytes 0. An adult male is about how tall? A) 75cm B) 75m C) 75mm The correct answer is A) since he is almost twice the standard height of a door knob, so almost meters, that is almost 00 cm.. Choose the most reasonable weight of a newborn baby: A).7g B)L C) 4kg D)0.4m The correct answer is C) since.7g is like a penny, while L measures volume and m is for length.
3 Conversion Tables The American system of measurements is based on the English system. American>American Capacity (volume) gallon = gal = 4 quarts = 4 qts quart = qt = pints = pt pint = pt = cups = c = 6 fluid ounces = 6 fl oz cup = c = 8 fluid ounces = 8 fl. oz (Note: a fluid ounce is not the same as the weight measure ounce.) fluid ounce = fl oz = tablespoons = tbsp tablespoon = tbsp = teaspoons = tsp Length foot = ft = inches = in mile = mi = 580 feet = 760 yd Mass pound = lb = 6 ounces = 6 oz ton = 000 pounds = 000 lb (This is sometimes called a short ton or ton(us). A long ton or ton(uk) is equal to 40 pounds.) Temperature Temperature is measured in degrees Fahrenheit ( F). Water freezes at F and boils at F SI> American meter =.096 yards =.808 feet = 9.7 inches kilometer =.67 miles gram =.057 ounces kilogram = 5.7 ounces =.05 pounds liter =.647 gallons =.0567 quarts C 5 9 F SI temperature measure is degree Celsius ( C). Water freezes at 0 C and boils at 00 F Another SI measure for temperature is Kelvin (K) where American> SI inch = meters =.54 centimeters yard = meters mile = meters = kilometers pound = grams ounce 8.5 grams gallon.784 liters quart.9464 liters fluid ounce.0957 liters = 9.57 milliliters tsp = 4.9 ml Everywhere in the World Time minute = min = 60 seconds = 60 s hour = hr = 60 min day = 4 hr Small amount of liquid Milliliters are used extensively in the sciences and medicine. The abbreviation cc is often used instead of ml. The relation of ml to volume is given by
4 Areas and volumes 4 Recall that area is measured in squares. One square foot or ft represents the area covered by a square foot by foot. Example: square feet = ft means squares foot by foot. The measure of the area of a rectangle ft by 4 ft is ft since it can be covered (tiled) with squares each foot by foot Volume is measured in cubes. One cubic centimeter or cm represents the volume that can be covered by a cube cm by cm by cm. Example: Find the volume of a box cm by cm by 4 cm. The question is, how many cubes cm on a side can be put into this box. cm cm 4 cm = 6 cm. That is, the box will hold 6 cubes cm by cm by cm. A conversion table could be made for areas and another could be made for volumes. The tables would include facts like ft = 44 in, yd = 9 ft, m = 0000 cm, ft = 78 in, etc. Fortunately, as shown below, this is not necessary. This handout will cover one method commonly used in the sciences to convert units. This method uses multiples of " in a convenient form. Examples: a) 4000 in =? ft Notice that ft = in. We write our " as ft so that in in 4000 in will cancel. in 4, 000in ft 4, 000 4, 000in ft. ft. notice that the in canceled in b) 8 ft =? in 8 ft in 8 ft 6in ft Notice that ft = in. This time we write our as in / ft so that the ft cancel leaving only in as the unit. c). mi =? in From the conversion tables on page notice that mi = 580 ft and ft = in. So we should write our in two different ways = 580ft mi and as = in. Using the first fraction we can ft convert miles into feet, while using the second fraction we can further convert the feet into inches. Combining these two we have the following:
5 .mi 580 ft in (.)(580)().mi in 4578in mi ft notice that the mi and the ft cancel 5 d) cm =? km Just like in the previous example we don t have a direct conversion between centimeters (cm) and kilometers (km). However, both are connected through meters (m). Therefore, we will use m the in two different ways: = and =. Using both, we get the following: 00cm 000m 46000cm m km cm km.46km 00cm 000 m (00)(000) notice that the cm and the m cancel e),000,000 fl oz =? gal From the conversion tables on page, we see that pt = 6 fl oz, qt = pt and gal = 4qt. We have to combine all three of these conversions to go from fluid ounces to gallons: fl oz pt qt gal fl oz gal 78.5gal 6 fl oz pt 4 qt (6)()(4) notice that the fl oz, the pt, and the qt cancel f) 8 ft =? yd The following is the standard method for handling areas and volumes. The conversion tables on page say that yd = ft. For square units, we have to use this conversion twice so that ft " cancels with ft ft. 8 ft yd yd 8 8 ft yd yd ft ft ()() Another way to obtain the same result is to create a new conversion formula for our specific prupose. Since yd = ft we immediately have yd = ft ft = 9ft and 8 ft 8 ft yd 8 ft yd 9 ft g),000,000 mm =? km This time we will use two conversions m = 000 mm and km = 000 m. Each of these two conversion will be used twice, since we are using square units: mm m m mm m 000mm 000mm Then we continue: km km m m km. Therefore mm = km. 000m 000m notice that mm " cancels with mm mm in the first row, and m cancels with m m in the second row of calculations above. h) 8 ft =? yd When dealing with cubic units, we have to apply each conversion formula three times. For example, since yd = ft, we can apply this conversion three times so that ft cancels with ft ft ft. 8 ft yd yd yd 8 8 ft yd yd ft ft ft ()()() km
6 Or, as in the example f) above, we can create our own conversion formula yd ft ft ft 7 ft and use it directly: 8 ft 8 ft yd 8 8 ft yd yd 7 ft 7 6 i) 8 ml =? cc This is a straight forward conversion, since cc = ml: 8mL cc 8mL 8cc ml j) L =? cc This time we have to use two conversions in a row: L = 000 ml and ml = cc: L 000mL cc ()(000) L cc 000cc L ml k) m =? cc For cubic units, we use conversion formulas three times: m 00cm 00cm 00cm cc (00)(00)(00) m cc cc m m m cm Alternatively, we can derive our own conversion formula for cubic units. Using m = 00cm, we get. m 00cm 00cm 00cm cm l) 74 F =? C C = 9 5 (F ) C = 9 5 (74 ) C. C m) 9 C =? F F = 5 9 C + F = [ 5 9 (9) + ] F = 84. F Compound Measures We now look at what is sometimes called compound measures: miles per hour = mi/hr, ft/sec, lb/ft, g/cm etc. Examples a) If a car is traveling 90 mi/hr, how many feet will the car travel in sec? In 5 sec? A direct way to convert these units is: 90mi 580 ft hr min (90)(580) 90 mi / hr ft / sec ft / sec hr mi 60 min 60sec (60)(60)
7 Alternatively, we can derive our own conversion formulas separately for miles and for hours: mi = 580ft and hr = 60min = 60 60sec = 600sec. We then replace miles with 580feet and hours with 600sec. Therefore 90 mi ft ft / sec hr 600sec 7 In second the car will travel ft. The car will travel (5 sec )( ft/sec) = 660 ft. in 5 seconds. b) A 90 lb weight is applied to an area /4 in by /4 inch. This pressure is equivalent to how many tons per ft? We have 90 pounds per (/4)(/4) in. We have [90 lb/(/6) in ] = 440 lb/in. 440lb ton in in (440)()() 440 lb / in ton / ft 0.68 ton / ft in 000lb ft ft 000 c) If a machine produces 5 parts in an 8 hour day, how many minutes will it take to produce 0 parts? We will first get minutes per part. 8hours 60 min 8(60) min/ part min/ part 5 parts hour 5 5 The time needed for 0 parts would be (0)(/5) minutes = 64/ minutes = minutes. d) If a machine produces 5 parts in an 8 hour day, how many parts can it produce in 0 minutes? We will first get parts per minute. Notice that we have parts in the numerator since we want parts in the final answer. 5 parts hour 5 5 parts / min parts / min 8hours 60 min 8(60) 5 In 0 minutes, the machine would produce 0 parts = 75 6 parts =. 4 6 e) Water is pouring into a plastic box m by m by. m at the rate of 5 ml per minute. How long will it take in hours to fill this box? The volume of the box is m m. m =. m. In minute, 5 ml will drip into the box. We will convert ml to cm to m and minutes to hours. min (00)(00)(00) ml cm cm cm hr hr. hr 5mL cm m m m 60 min (5)(60) m m. hrs for each m. It will take (.)(.) hr = 40 hr to fill the box Homework:. 000 yd =? in. 000 in =? ft. 000 in =? yd ft =? in yd =? in in =? ft in =? yd ft =? in yd =? in in =? ft. 000 in =? yd. 000 ft =? in
8 .,000,000 mm =? cm. 50 mi =? yd lb/in =? tons/ft 8 4.,000,000 mg =? cg. 50 mi =? in tons / ft =? lb / in 5.,000,000 mm =? m.,000,000 tons =? oz km/hr =? m/sec 6.,000,000 μg =? g 4.,000,000 oz =? tons m/min =? km / hr 7.,000,000 mg =? g 5.,000,000 fl. oz. =? gal m/sec =? km/hr 8.,000,000 cm =? m 6.,000 fl. oz. =? quarts km / hr =? m / min 9.,000,000 ml =? L 7. 4 F =? C 55. cc/sec =? L/hr 0.,000,000 μm =? cm 8. 7 F =? C 56. L / hr =? cc / min.,000,000 mm =? km 9. C =? F 57. m =? L.,000,000 cm =? km C =? F L =? m.,000,000 mg =? kg 4. 5 C =? K m =? ft 4.,000,000 g =? mg K =? C ft =? m 5.,000,000 km =? m ft/sec =? mi/hr 6. 9 qt =? m 6.,000,000 kg =? g mi / hr =? ft / sec. 6. m =? qt 7.,000,000 km =? cm gal / min =? gal/hr 6.,000 kg=? lbs 8.,000,000 ml =? L pints / min =? gal / hr lbs =? kg 9.,000,000 kg =? mg 0.,000,000 cg =? g 47. g/cm =? kg/m kg / m =? g / mm 65.. mg / mm =? oz / in oz / in =? mg / mm 67. A car traveling at a constant speed travels 75 miles in 4 hours. How many minutes will it take for the car to travel 000 feet? 68. A car traveling at a constant speed travels 000 ft in seconds. In miles, how far will this car travel in 4 hours? 69. A car traveling at a constant speed travels 75 miles in 4 hours. How many feet will the car travel in 0 minutes? 70. A car traveling at a constant speed travels 000 ft in seconds. How long in hours will it take to travel 75 miles? 7. If a car is traveling 75 miles per hour, how many feet will it travel in 0 seconds? 7. If a car is traveling 75 kilometers per hour, how many meters will it travel in 0 seconds? 7. A leaky faucet drips at the rate of pint per hour. How many gallons will drip in a 4 hour day? 74. A leaky faucet drips at the rate of 00 ml per hour. How many liters will drip in a 4 hour day?
9 75. A leaky faucet drips at the rate of pint per hour. How long will it take to fill a 0 gallon container? 76. A leaky faucet drips at the rate of 00 ml per hour. How long will it take to fill a 0 liter container? 77. You wish to give a patient 5 liters of fluid intravenously in 4 hours. If this is done at a constant rate, how many cc must be given per minute? 78. You wish to give a patient 4 liters of fluid in 8 hours. If this is done at a constant rate, how many cc must be given per minute? 79. You are giving a patient.5 cc / min of a given solution intravenously. If this continues for 8 hours, how many liters of this solution will you need? 80. You are giving a patient cc / min of a given solution intravenously. If this is done for 4 hours, how many liters of this solution will you need? 8. A recipe calls for 00 ml water. In the English system, how much water is this? 8. A recipe calls for cups water. In the metric system, how much water is this? 8. A car is traveling 00 km / hr, How many feet does this car travel every second? 84. A car is traveling 00 miles / hour. How many meters does this car travel every second? 85. John has a bicycle whose tires are 7 inches in diameter. If John rides his bike mile, how many revolutions does each tire make? A vehicle has a wheel 5 inches in diameter. If the vehicle travels miles, how many revolutions does the wheel make? inches is how many meters? millimeters is how many inches? 89. A large block of butter measures. meters by.4 meters by 5 meters. How many tablespoons of butter are in this block? 90. You wish to form a cube formed from,000,000 tablespoons of butter. What is the length of a side in meters? feet? inches? (Recall: a cube has length = width = height.) 9. An automatic machine produces part every. minutes. How many parts are produced in 8 hours if the machine runs constantly? 9. A machine produces 4500 parts in an 8 hour day. In minutes, how long will it take to produce 00 parts? 9. An automatic machine produces part every. minutes. How long in hours will it take to produce 5000 parts? 94. If a machine produces 45 parts per hour. how many minutes will it take to produce each part? 95. A certain medication recommends that each day a person be given 8 mg for every 00 lbs of weight. John weighs 8 lbs. If his measurements are exact, how many kg of this medication will he take in 65 days? 96. A certain medication recommends that each day a person be given 8 mg for every 00 lbs of weight. If 0,000 people take this medication each day and their average weight is 80 lbs, how many kilograms of this drug are used daily?
10 Solutions: 0 000yd ft in. 000 yd (000)()() in 6000in yd ft 000in ft yd in yd 7 yd in ft ()() yd ft ft in in (000)()()()() in 96000in yd yd ft ft 000in ft ft yd yd yd.77 yd in in ft ft ()()()() 000 yd ft ft ft in in in 9. (000)()()()()()() in in yd yd yd ft ft ft. 000in ft ft ft yd yd yd 000 yd.0yd in in in ft ft ft ()()()()()() mm m 00 cm (000000)(00) cm 00000cm 000mm m mm m m 000m 000mm mg g g 000g 000mg ml L L 000L 000mL mm m km km km 000 mm 000 m (000)(000) mg g kg kg kg 000mg 000 g (000)(000) km 000 m (000000)(000) m m km km 000m 00cm 7. (000000)(000)(00) cm cm 0 cm km m kg 000 g (000000)(000) g g kg. 50 mi 580 ft yd (50)(580) yd 88000yd mi ft
11 000000tons 000lb 6oz 0. (000000)(000)(6) oz oz. 0 oz ton lb fl oz pt qt gal gal 78.5gal 6 fl oz pt 4 qt (6)()(4) 7. C = 5 9 (F ) ; C = 5 9 (4 ) C 5.6 C 9. F = 9 5 C + ; F = 9 5 () + F = 5.6 F 4. K = C + 7.5; K = K = 98.5 K ft mi 60sec 60min (00)(60)(60) mi 68.8 mi / hr gal 60min (5)(60) gal / hr 00 gal / hr sec 580 ft min hr 580hr min hr g kg 00cm 00cm 00 cm ()(00)(00)(00) 47. kg / m 000 kg / m cm 000g m m m lb ton in in (00)()() 49. tons / ft 4.4 tons / ft in 000lb ft ft km 000 m hr min (00)(000) m/ sec 7.78 m/ sec hr km 60min 60sec (60)(60) m km 60sec 60min (00)(60)(60) km / hr 60 km / hr sec 000m min hr cc ml L 60sec 60min ()(60)(60) L / hr 7. L / hr sec cc 000ml min hr 000 m 00 cm 00 cm 00 cm ml L ()(00)(00)(00) m m m cm 000 ml 000 L 000 L 4m 9.7 in 9.7 in ft ft 4(9.7)(9.7) ft 4.05 ft m m in in ()() 9qt L 000 ml cm m m m 9(000) m.0567 qt L ml 00cm 00cm 00 cm (.0567)(00)(00)(00) ,000 kg. lbs,00lbs kg 65.. mg g kg lbs oz mm mm mm m m m.85 oz mm 000 mg 000 g kg lb m m m 9.7 in 9.7 in 9.7 in in 4hours mile 60min (4)(60) 67. min/ ft min/ ft ; 75miles 580 ft hour (75)(580) The car will take approximately.6 minutes to travel 000 feet miles 580 ft hour (75)(580) ft / min 850 ft / min ; 4hours mile 60 min (4)(60)
12 In 0 minutes the car would travel (0)(850) ft = 8500 ft mi 580 ft hr min (75)(580) ft 0 ft hr mi 60 min 60 sec (60)(60) sec sec In 0 seconds the car will travel (0)(0) ft. = 00 ft. 7. pt 4 4 qt gal hours gal / day gal hour pt 4qt day ()(4) day 75. hour pt 4 qt 8 hours / gal ; It would take (0)(8) hours = 80 hours to fill a 0 gallon container. pt qt gal 77. 5L hour 000mL cc (5)(000) cc / min.47 cc / min 4hours 60min L ml (4)(60) cc 60min ml L (.5)(60) L / hour.5 L / hour.you need (8)(.5) L =. L in 8 hours. min hour cc 000mL ml pt cups 8. pt pt, also pt 4 cups and 00 ml Tbs 0Tbs 480 ml pt 0mL 00 km.64 miles 580 ft hr min (00)(.64)(580) ft hr km mile 60min 60sec (60)(60)sec 8. ft 9.4 sec 85. C = π d = π r ; C = π 7 in. ; revolution = rev = circumference = 7 π in. ; mile 580 (580)() 747 ft in rev rev rev mile ft 7in in m 000 m 5.4in 9.7in Volume =. m x.4 m x 5 m = 9. m ; 9.m 00cm 00cm 00cm ml tbsp (9.)(00)(00)(00) tbsp,600,000 tbsp m m m cm 5mL 5 9. part 60 min 60 parts 6. parts (8)(6.) parts ; 08.8 parts in 8 hours.min hr. hr hour 8hours 9..min. (.08)(5000) hr hr.08 hr hrs ; part 60min 60 part part 5000 parts It will take approximately 9 hours to produce 5000 parts mg 8lbs 9.04 mg 00lbs each day; 9.04mg each day x 65 days = mg mg g kg kg kg 000mg 000 g (000)(000) In one year, John would take approximately.007 kg.
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