#set document(title: "9.4 Measuring Weight", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.4#h(0.6em)Measuring Weight #figure(figph[A weighing scale.], alt: "A weighing scale.", caption: [Weight scale at the local Antigua market]) === Learning Objectives After completing this section, you should be able to: + Identify reasonable values for weight applications. + Convert units of measures of weight. + Solve application problems involving weight. In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car weighs about 2 metric tons. Any discussion about metric weight must also include a conversation about #strong[mass]. Scientifically, mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects with the same mass will weigh less on the moon than on Earth. Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well. === Reasonable Values for Weight To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation. #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Metric_System:_units_of_weight")[Metric System: Units of Weight] ] #examplebox("Example 1")[Identifying Reasonable Units for Weight][ Which is the more reasonable value for the weight of a newborn baby: - 3.5 kg or - 3.5 g? #solutionbox[ Using our reference weights, a baby weighs more than 3.5 paperclips, so 3.5 kilograms is a more reasonable value for the weight of a newborn baby. ] ] #examplebox("Example 2")[Determining Reasonable Values for Weight][ Which of the following represents a reasonable value for the weight of three lemons? - 250 g, - 2,500 g, or - 250 kg? #solutionbox[ Because a kilogram is about 2.2 pounds, we can eliminate 250 kg as it is way too heavy. 2,500 grams is equivalent to 2.5 kilograms, or about five pounds, which is again, too heavy. So, a reasonable value for the weight of three lemons would be 250 grams. ] ] #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[ #emph[How Do You Measure the Weight of a Whale?] It is impossible to weigh a living whale. Fredrik Christiansen from the Aarhus Institute of Advanced Studies in Denmark developed an innovative way to measure the weight of whales. Using images taken from a drone and computer modeling, the weight of a whale can be estimated with great accuracy. ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Using_drones_to_weigh_whales?")[Using Drones to Weigh Whales?] ] #examplebox("Example 3")[Explaining Reasonable Values for Weight][ The blue whale is the largest living mammal on Earth. Which of the following is a reasonable value for the weight of a blue whale: 149 g, 149 kg, or 149 mt? Explain your answer. #solutionbox[ A reasonable value for the weight of a blue whale is 149 metric tons. Both 149 g and 149 kg are much too small a value for the largest living mammal on Earth. ] ] === Converting Like Units of Measures for Weight Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in . #figure(figph[An illustration shows three units: milligram, gram, and kilogram.], alt: "An illustration shows three units: milligram, gram, and kilogram.", caption: [Common Conversion Factors for Metric Weight Units]) #examplebox("Example 4")[Converting Metric Units of Weight Using Multistep Division][ How many kilograms are in 24,300,000 milligrams? #solutionbox[ Use division to convert from a smaller metric weight unit to a larger metric weight unit. To convert from milligrams to kilograms, #strong[Step 1:] Divide the value of the weight in milligrams by 1,000 to first convert from milligrams to grams. #strong[Step 2:] Divide by 1,000 again to convert from grams to kilograms. #math.equation(block: true, alt: "the fraction 24,300,000 ⁢ mg over 1,000, equals, 24,300 ⁢ g; the fraction 24,300 ⁢ g over 1,000, equals, 24.3 ⁢ kg")[$frac(24","300","000 "mg", 1","000) & = & 24,300 upright(g) \ frac(24","300 upright(g), 1","000) & = & 24.3 "kg"$] So, 24,300,000 milligrams are equivalent to 24.3 kilograms. ] ] #examplebox("Example 5")[Converting Metric Units of Weight Using Multiplication][ The average ostrich weighs approximately 127 kilograms. How many grams does an ostrich weigh? #solutionbox[ Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms to grams, multiply the value of the weight by 1,000. #math.equation(block: true, alt: "127 ⁢ kg times 1,000 equals 127,000 ⁢ g")[$127 "kg" × 1,000 = 127,000 upright(g)$] The average ostrich weighs 127,000 grams. ] ] #examplebox("Example 6")[Converting Metric Units of Weight Using Multistep Multiplication][ How many milligrams are there in 0.025 kilograms? #solutionbox[ Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms to grams, #strong[Step 1:] Multiply the value of the weight by 1,000. #strong[Step 2:] Multiply the result by 1,000 to convert from grams to milligrams. #math.equation(block: true, alt: "0.025 ⁢ kg times 1,000 equals 25 ⁢ g; 25 g times 1,000 equals 25,000 mg")[$0.025 "kg" × 1,000 = 25 upright(g) \ 25 #h(0.28em) upright(g) × 1,000 = 25,000 #h(0.28em) "mg"$] So, 0.025 kilograms is equivalent to 25,000 milligrams. ] ] #notebox("Video", rgb("#DC2626"), rgb("#DC2626"), rgb("#f7f8fa"))[ #link("https://openstax.org/r/Convert_mg-g-kg")[Metric Units of Mass: Convert mg, g, and kg] ] === Solving Application Problems Involving Weight From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating. #examplebox("Example 7")[Comparing Weights to Solve Problems][ The maximum weight for a child to safely use a car seat is 29 kilograms. If a child weighs 23,700 grams, can the child safely use the car seat? #solutionbox[ #strong[Step 1:] Convert the child’s weight in grams to kilograms. #math.equation(block: true, alt: "23,700 ⁢ g divided by 1,000 equals 23.7 ⁢ kg")[$23,700 upright(g) ÷ 1,000 = 23.7 "kg"$] #strong[Step 2:] Compare the two weights. #math.equation(block: true, alt: "23.7 ⁢ kg less than 29 ⁢ kg")[$23.7 "kg" < 29 "kg"$] Yes, the child can safely use the car seat. ] ] #examplebox("Example 8")[Solving Multistep Weight Problems][ A recipe for scones calls for 350 grams of flour. How many kilograms of flour are required to make 4 batches of scones? #solutionbox[ #strong[Step 1:] Multiply the grams of flour need by 4 to determine the total amount of flour needed. #math.equation(block: true, alt: "350 ⁢ g times 4 equals 1,400 ⁢ g")[$350 upright(g) × 4 = 1,400 upright(g)$] #strong[Step 2:] Convert from grams to kilograms. #math.equation(block: true, alt: "the fraction 1,400 ⁢ g over 1,000 equals 1.4 ⁢ kg")[$frac(1","400 upright(g), 1","000) = 1.4 "kg"$] So, 1.4 kilograms of flour are needed to make four batches of scones. ] ] #examplebox("Example 9")[Solving Complex Weight Problems][ The average tomato weighs 140 grams. A farmer needs to order boxes to pack and ship their tomatoes to local grocery stores. They estimate that this year’s harvest will yield 125,000 tomatoes. A box can hold 12 kilograms of tomatoes. How many boxes does the farmer need? #solutionbox[ #strong[Step 1:] Determine the total estimated weight of the harvested tomatoes. #math.equation(block: true, alt: "140 ⁢ g times 125,000 equals 17,500,000 g")[$140 upright(g) × 125,000 = 17,500,000 #h(0.28em) upright(g)$] #strong[Step 2:] Convert the total weight from grams to kilograms. #math.equation(block: true, alt: "17,500,000 ⁢ g divided by 1,000 equals 17,500 ⁢ kg")[$17,500,000 upright(g) ÷ 1,000 = 17,500 "kg"$] #strong[Step 3:] Divide the weight of the tomatoes by the weight each box can hold. #math.equation(block: true, alt: "17,500 ⁢ kg divided by 12 ⁢ kg approximately equals 1,458")[$17,500 "kg" ÷ 12 "kg" ≈ 1,458$] So, the farmer will need to order 1,458 boxes. ] ] === Key Terms - mass === Key Concepts - Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of an object never changes; the weight of an object changes depending on the force of gravity. An object with the same mass would weigh less on the moon than on Earth because the moon’s gravity is less than that of Earth. - In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or kilograms. === Videos - #link("https://openstax.org/r/Metric_System:_units_of_weight")[Metric System: Units of Weight] - #link("https://openstax.org/r/Using_drones_to_weigh_whales?")[Using Drones to Weigh Whales?] - #link("https://openstax.org/r/Convert_mg-g-kg")[Metric Units of Mass: Convert mg, g, and kg] === Formulas You can convert between metric weight units using the conversion factors shown.