# EXAMPLE EXERCISE 3.1 Metric Basic Units and Prefixes

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1 EXAMPLE EXERCISE 3.1 Metric Basic Units and Prefixes Give the symbol for each of the following metric units and state the quantity measured by each unit: (a) gigameter (b) kilogram (c) centiliter (d) microsecond We compose the symbol for each unit by combining the prefix symbol and the basic unit symbol. If we refer to Tables 3.1 and 3.2, we have the following: (a) Gm, length (b) kg, mass (c) cl, volume (d) µs, time

2 EXAMPLE EXERCISE 3.1 Metric Basic Units and Prefixes Continued Give the symbol for each of the following metric units and state the quantity measured by each unit: (a) nanosecond (b) milliliter (c) decigram (d) megameter Answers: (a) ns, time; (b) ml, volume; (c) dg, mass; (d) Mm, length What is the basic unit of length, mass, and volume in the metric system?

3 EXAMPLE EXERCISE 3.2 Metric Unit Equations Complete the unit equation for each of the following exact metric equivalents: (a) 1 Mm =? m (b) 1 kg =? g (c) 1 L =? dl (d) 1 s =? ns We can refer to Table 3.2 as necessary. (a) The prefix mega- means 1,000,000 basic units; thus, 1 Mm = 1,000,000 m. (b) The prefix kilo- means 1000 basic units; thus, 1 kg = 1000 g. (c) The prefix deci- means 0.1 of a basic unit, thus, 1 L = 10 dl. (d) The prefix nano- means of a basic unit, thus; 1 s = 1,000,000,000 ns. Alternatively, we can express the unit equation using exponential numbers: (a) 1 Mm = m (b) 1 kg = g (c) 1 L = dl (d) 1 s = ns

4 EXAMPLE EXERCISE 3.2 Metric Unit Equations Continued Complete the unit equation for each of the following exact metric equivalents: (a) 1 Gm =? m (b) 1 g =? cg (c) 1 L =? µl (d) 1 s =? ms Answers: (a) 1 Gm = m (b) 1 g = cg (c) 1 L = µl (d) 1 s = ms How many significant digits are in the following unit equation? 1 m = 1000 mm

5 EXAMPLE EXERCISE 3.3 Metric Unit Factors Write two unit factors for each of the following metric relationships: (a) kilometers and meters (b) grams and decigrams We start by writing the unit equation to generate the two unit factors. (a) The prefix kilo- means 1000 basic units; thus, 1 km = 1000 m. The two unit factors are (b) The prefix deci- means 0.1 basic unit; thus, 1 g = 10 dg. The two unit factors are Write two unit factors for each of the following metric relationships: (a) liters and milliliters (b) megaseconds and seconds Answers: (a) 1 L/1000 ml and 1000 ml/1 L (b) 1 Ms/1,000,000 s and 1,000,000 s/1 Ms How many significant digits are in the following unit factor? 1 L/1000 ml

6 EXAMPLE EXERCISE 3.4 Metric Metric Conversion A hospital has 125 deciliter bags of blood plasma. What is the volume of plasma expressed in milliliters? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Given that 1 L = 10 dl, and 1 L = 1000 ml, the two pairs of unit factors are Unit Analysis Map

7 EXAMPLE EXERCISE 3.4 Metric Metric Conversion Continued We apply the unit factor 1 L/10 dl to cancel deciliters, and 1000 ml/1 L to cancel liters. The given value, 125 dl, limits the answer to three significant digits. Since each unit factor is derived from an exact equivalent, neither affects the number of significant digits in the answer. A dermatology patient is treated with ultraviolet light having a wavelength of 375 nm. What is the wavelength expressed in centimeters? Answer: cm ( cm) Express the volume of a cube 1 cm on a side in milliliters.

8 EXAMPLE EXERCISE 3.5 Metric Metric Conversion The mass of Earth is kg. What is the mass expressed in megagrams? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Given that 1 kg = 1000 g, and 1 Mg = 1,000,000 g, the two pairs of unit factors are Unit Analysis Map Earth The mass of Earth is about 6,000,000,000,000,000,000,000,00 0 kilograms.

9 EXAMPLE EXERCISE 3.5 Metric Metric Conversion Continued We apply the unit factor 1000 g/1 kg to cancel kilograms, and 1 Mg/1,000,000 g to cancel grams. The given value limits the answer to three significant digits. Since each unit factor is derived from an exact equivalent, neither affects the number of significant digits in the answer. Light travels through the universe at a velocity of cm/s. How many gigameters does light travel in one second? Answer: Gm ( Gm) How many significant digits are in the following unit factor? 1 g/1000 mgn

10 EXAMPLE EXERCISE 3.6 Metric English Conversion A half-gallon carton contains 64.0 fl oz of milk. How many milliliters of milk are in a carton (given that 1 qt = 32 fl oz)? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Given that 1 qt = 32 fl oz and that in Table 3.3 we find that 1 qt = 946 ml. The two pairs of unit factors are Unit Analysis Map

11 EXAMPLE EXERCISE 3.6 Metric English Conversion Continued We apply the unit factor 1 qt/32 fl oz to cancel fluid ounces 946 ml/1 qt to cancel quarts., and The given value, and unit factor 2, each limits the answer to three significant digits. Since unit factor 1 is derived from an exact equivalent, it does not affect the number of significant digits in the answer. A plastic bottle contains 5.00 gallons of distilled water. How many liters of distilled water are in the bottle (given that 1 gal = 4 qt)? Answer: 18.9 L How many significant digits are in the following unit factor? 1 qt/946 ml Milk Carton A half-gallon carton of milk has a volume of 64.0 fluid ounces.

12 EXAMPLE EXERCISE 3.7 Metric English Conversion If a tennis ball weighs 2.0 oz, what is the mass of the tennis ball in grams? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Given that 1 lb = 16 oz and that in Table 3.3 we find that 1 lb = 454 g. The two pairs of unit factors are Unit Analysis Map

13 EXAMPLE EXERCISE 3.7 Metric English Conversion Continued We apply the unit factor 1 lb/16 oz to cancel ounces to cancel pounds., and 454 g/1 lb The given value, 2.0 oz, limits the answer to two significant digits. Unit factor 1 has no effect as it is derived from an exact equivalent, and unit factor 2 has three significant digits. If a tennis ball has a diameter of 2.5 inches, what is the diameter in millimeters? Answer: 64 mm How many significant digits are in the following unit factor? Tennis Balls A tennis ball weighs about 2.0 ounces, and has a diameter of 2.5 inches. 1 kg/2.20 lb

14 EXAMPLE EXERCISE 3.8 Conversion of a Unit Ratio If a Mazda Miata is traveling at 95 km/h, what is the speed in meters per second (given that 1 km = 1000 m, and 1 h = 3600 s)? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Given that 1 km = 1000 m, and 1 h = 3600 s, the two pairs of unit factors are Unit Analysis Map

15 EXAMPLE EXERCISE 3.8 Conversion of a Unit Ratio Continued We apply the unit factor 1000 m/1 km to cancel kilometers 1 h/3600 s to cancel hours., and The given value has two significant digits, so the answer is limited to two digits. Since each unit factor is derived from an exact equivalent, neither affects the number of significant digits in the answer. If a runner completes a 10K race in minutes (min), what is the 10.0 km pace in miles per hour (given that 1 mi = 1.61 km)? Answer: 11.5 mi/h Mazda Miata A Mazda Miata. Which speed is faster: 65 mi/h or 65 km/h?

16 EXAMPLE EXERCISE 3.9 Volume Calculation for a Rectangular Solid If a stainless steel rectangular solid measures 5.55 cm long, 3.75 cm wide, and 2.25 cm thick, what is the volume in cubic centimeters? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. We can calculate the volume of the rectangular solid by multiplying length times width times thickness: l w t cm 3.75 cm 2.25 cm = 46.8 cm 3 The answer is rounded off to three significant digits because each given value has three significant digits.

17 EXAMPLE EXERCISE 3.9 Volume Calculation for a Rectangular Solid Continued If a rectangular brass solid measures 52.0 mm by 25.0 mm by 15.0 mm, what is the volume in cubic millimeters? Answer: mm 3 ( mm 3 ) Express the volume of a cube 10 cm on a side in liters.

18 EXAMPLE EXERCISE 3.10 Thickness Calculation for a Rectangular Solid A sheet of aluminum foil measures 25.0 mm by 10.0 mm, and the volume is 3.75 mm 3. What is the thickness of the foil in millimeters? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. We can calculate the thickness of the foil by dividing the volume by length and width. Since the unit of volume is mm 3, we obtain the thickness in mm by unit cancellation. The answer is rounded off to three significant digits because each given value has three significant digits.

19 EXAMPLE EXERCISE 3.10 Thickness Calculation for a Rectangular Solid Continued A sheet of aluminum foil measures 35.0 cm by 25.0 cm, and the volume is 1.36 cm 3. What is the thickness of the foil in centimeters? Answer: cm ( cm) Which of the following is the greatest thickness? 1 mm, 0.1 cm, or m Aluminum foil A thin sheet of aluminum foil.

20 EXAMPLE EXERCISE 3.11 Metric English Volume Conversion Given that an automobile engine has a volume displacement of 498 cm 3 in each cylinder, express the volume in cubic inches. Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Unit Analysis Map

21 EXAMPLE EXERCISE 3.11 Metric English Volume Conversion Continued We can calculate the volume in cubic inches by converting the volume given in cubic centimeters. Notice that cm units do not cancel cm 3. To obtain cubic inches, we must apply the unit factor, 1 in./2.54 cm, three times. Thus, The given value, and unit factors, each limits the answer to three significant digits. Given that an SUV has a 244 in. 3 engine, express the engine volume in liters. Answer: 4.00 L Which of the following is the greater volume? 500 ml or 500 cm 3 Jonathan Player/The New York Times Mini Cooper The Mini Cooper is built in England and marketed in the United States by BMW.

22 EXAMPLE EXERCISE 3.12 Volume by Displacement A quartz stone weighing g is dropped into a graduated cylinder. If the water level increases from 25.0 ml to 36.5 ml, what is the volume of the quartz stone? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. We can calculate the displaced volume in milliliters by subtracting the initial volume from the final volume. Answer: 55.5 ml 36.5 ml 25.0 ml = 11.5 ml Hydrogen peroxide decomposes to give oxygen gas, which displaces a volume of water into a beaker. If the water level in the beaker increases from 50.0 ml to ml, what is the volume of oxygen gas?

23 EXAMPLE EXERCISE 3.12 Volume by Displacement Continued Which of the following has the greater volume? 1 ml or 1 cm 3

24 EXAMPLE EXERCISE 3.13 Density Calculation If a platinum nugget has a mass of g and a volume 10.0 cm 3, what is the density of the metal? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. We can calculate the density of the platinum nugget by comparing the mass of metal, g, to its volume, 10.0 cm 3. The given volume has three significant digits, so the answer is rounded off to three digits. It is interesting to note that platinum metal is more dense than lead (d = 11.3 g/cm 3 ), and more valuable than gold. Platinum Nugget A small nugget of precious platinum metal.

25 EXAMPLE EXERCISE 3.13 Density Calculation Continued Carbon tetrachloride is a solvent used for degreasing electronic parts. If 25.0 ml of carbon tetrachloride has a mass of g, what is the density of the liquid? Answer: 1.59 g/ml Which of the following has the greater density: ice or water?

26 EXAMPLE EXERCISE 3.14 Density as a Unit Factor An automobile battery contains 1275 ml of sulfuric acid. If the density of battery acid is 1.84 g/ml, how many grams of acid are in the battery? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? From the definition of density, 1.84 g = 1 ml; thus, the two unit factors are 1.84 g/1 ml, and its reciprocal 1 ml /1.84 g. Unit Analysis Map

27 EXAMPLE EXERCISE 3.14 Density as a Unit Factor Continued We apply the unit factor 1.84 g/1 ml to cancel milliliters in the denominator., which appears The given value, 1275 ml, has four significant digits, but the unit factor has only three digits. Therefore, the answer is rounded off to three digits. Battery acid is the common name for sulfuric acid, which annually ranks as the most important industrial chemical. The most abundant gases in our atmosphere are nitrogen, oxygen, and argon. What is the volume of 1.00 kg of air? (Assume the density of air is 1.29 g/l.) Answer: 775 L Which of the following is the greater density? 1 g/ml or 1 kg/l Automobile Battery An automobile battery contains lead plates and is filled with sulfuric acid.

28 EXAMPLE EXERCISE 3.15 Density as a Unit Factor A 1.00-in. cube of copper measures 2.54 cm on a side. What is the mass of the copper cube (given that d of copper = 8.96 g/cm 3 )? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? From the definition of density, 8.96 g = 1 cm 3 ; thus, the two unit factors are 8.96 g/1 cm 3, and its reciprocal 1 cm3/8.96 g. Unit Analysis Map

29 EXAMPLE EXERCISE 3.15 Density as a Unit Factor Continued First, we find the volume of the copper cube. We obtain the volume of the cube, 16.4 cm 3, by multiplying (2.54 cm) (2.54 cm) (2.54 cm). We use the given density, 8.96 g/1 cm 3, as a unit factor to cancel cubic centimeters, which appears in the denominator. The given value and unit factor each has three significant digits, so the answer is rounded off to three significant digits. A cube of silver is 5.00 cm on a side and has a mass of g. What is the density of silver? Answer:10.5 g/cm 3 Copper Metal A 1.00-inch cube of copper metal. If some humans float in water and other sink, what is the approximate density of the human body?

30 EXAMPLE EXERCISE 3.16 F and C Temperature Conversions Normal body temperature is 98.6 F. What is normal body temperature in degrees Celsius? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. To calculate C, we refer to Figure 3.6 and compare the Celsius and Fahrenheit temperature scales. The conversion from F to C is as follows. Simplifying and canceling units gives The given value, 98.6 F, has three significant digits, so the answer is rounded off to three digits. Since 32 F and 100 C/180 F are exact numbers, neither affects the significant digits in the answer. Figure 3.6 Temperature Scales A Fahrenheit, a Celsius, and a Kelvin thermometer are placed in (a) ice water and (b) boiling water. Notice the freezing point and boiling point on each scale. The number of divisions is 180 units on the Fahrenheit scale, 100 units on the Celsius scale, and 100 units on the Kelvin scale.

31 EXAMPLE EXERCISE 3.16 F and C Temperature Conversions Continued The average surface temperature of Mars is 55 C. What is the average temperature in degrees Fahrenheit? Answer: 67 F What is the relationship between the Celsius and centigrade temperature scales? Australian Stamp The cartoon illustrates that 38 C is approximately equal to 100 F

32 EXAMPLE EXERCISE 3.17 C and K Temperature Conversions Dermatologists use liquid nitrogen to freeze skin tissue. If the Celsius temperature of liquid nitrogen is 196 C, what is the Kelvin temperature? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? No unit factor is required. Given the Celsius temperature, we add 273 units to find the corresponding Kelvin temperature C = 77 k Liquid Nitrogen Although nitrogen is normally a gas, it liquefies at 196 C. When liquid nitrogen is poured from a Thermos, it is cold enough to freeze the moisture in air and form a white mist.

33 EXAMPLE EXERCISE 3.17 C and K Temperature Conversions Continued The secret to fire-walking is to first walk barefoot through damp grass and then step lively on the red-hot coals. If the bed of coals is 1475 K, what is the Celsius temperature? Answer: 1202 C Which of the following temperatures does not exist? 100 F, 100 C, 100 K

34 EXAMPLE EXERCISE 3.18 Energy Conversion Burning one liter of natural gas produces 9.46 kcal of heat energy. Express the energy in kilojoules (given that 1 kcal = kj). Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Since the unit equation is 1 kcal = kj, the two unit factors are 1 kcal/4.184 kj, and its reciprocal kj /1 kcal. Unit Analysis Map

35 EXAMPLE EXERCISE 3.18 Energy Conversion Continued We apply the unit factor kj /1 kcal to cancel kilocalories appears in the denominator., which The given value has three significant digits, and the unit factor has four digits. Thus, we round off the answer to three significant digits. Burning one gram of gasoline produces 47.9 kj of energy. Express the heat energy in kilocalories (given that 1 kcal = kj). Answer: 11.4 kcal If an aerosol can feels cold after releasing the spray, is heat flowing from the can or from your hand? Bunsen Burner A laboratory burner that uses natural gas for fuel.

36 EXAMPLE EXERCISE 3.19 Specific Heat An energy-efficient home may have solar panels for heating water. If 350,000 cal heat water from 20.0 C to 35.0 C, what is the mass of water (specific heat = 1.00 cal/g C)? Strategy Plan Step 1: What unit is asked for in the answer? Step 2: What given value is related to the answer? Step 3: What unit factor(s) should we apply? Unit Analysis Map

37 EXAMPLE EXERCISE 3.19 Specific Heat Continued We apply the unit factor to cancel calories, and ( ) C to cancel degrees Celsius. A 725 g steel horseshoe is heated to 425 C and dropped into a bucket of cold water. If the horseshoe cools to 20 C and the specific heat of steel is 0.11 cal/g C, how much heat is released? Answer: 32,000 cal (32 kcal) If the crust of an apple pie cooks faster than the filling, which has the higher specific heat: the crust or apple filling? Solar Panels on Rooftop Solar panels collect light energy from the Sun, which is converted to heat energy for the home.

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