Measurement in Two Systems · 1.6 Describe and explain a personal strategy used to determine a...

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MeasurementinTwoSystems

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P a g e 2|MeasurementCopyrightMathbeacon.com.Usewithpermission.DonotuseafterJune2019.

MEASUREMENT TOPICS

Solveproblemsthatinvolvelinearmeasurementusing:

• SIandimperialunitsofmeasure.

• Estimationstrategies.• Measurementstrategies.

1.1Providereferentsforlinearmeasurement.

1.2CompareSIandImperialunitsofmeasure.

1.3Estimatealinearmeasureandexplaintheprocess.

1.4Justifythechoiceofunitsusedfordeterminingalinearmeasurement.

1.5Solveproblemsthatinvolvelinearmeasureusinginstrumentsavailable.

1.6Describeandexplainapersonalstrategyusedtodeterminealinearmeasurement.

ApplyproportionalreasoningtoproblemsthatinvolveconversionsbetweenSIandimperialmeasurement.

2.1UseproportionalreasoningtoconvertwithinorbetweenSIandImperialsystems.

2.2Solveaproblemthatrequiresconversionbetweenunits.

2.3Verifyusingunitanalysis,aconversionbetweenunits.

2.4Justify,usingmentalmathematics,thereasonablenessofasolutiontoaconversionproblem.

Solveproblems,usingSIandimperialmeasurement,thatinvolvethesurfaceareaandvolumeof3-Dobjects,including

• Rightcones• Rightcylinders• Rightprisms• Rightpyramids• Spheres.

3.1Sketchadiagramtorepresentaproblemthatinvolvessurfaceareaorvolume.

3.2DeterminetheSAofarightcone,rightprism,rightpyramid,orsphereusinganobjectordiagram.

3.3DeterminetheVofarightcone,rightprism,rightpyramid,orsphereusinganobjectordiagram.

3.4Determineanunknowndimensionofarightcone,rightprism,rightpyramid,orspheregiventheSAorVandremainingdimensions.

3.5SolveaproblemthatinvolvesSAorV.

3.6Comparetheformula/relationshipbetweenvolumesofrightconesandrightcylindersorrightprismsandrightpyramids.

[C]Communication[PS]ProblemSolving,[CN]Connections[R]Reasoning,[ME]MentalMathematics[T]Technology,andEstimation,[V]Visualization

Key Terms Term Definition

Linearmeasurement.

Dimension

Length

Height

Radius

Diameter

Perimeter

Linearfoot

Conversionfactor

Baseoftriangle

Heightofatriangle(altitude)

2-dimensional

SurfaceArea

Squarefoot.

Geometricnet

3-dimensional

Base(orbasearea)ofafigure

Term Definition

Volume

Cubiccentimetres(ccorcm3)

Millilitre

Gallon

Rectangle

Square

Parallelogram

Trapezoid

Circle

RectangularPrism(&Cube)

RightTriangularPrism

Pyramid(triangular,rectangular,square)

RightCone

RightCylinder

Sphere

Hemisphere

UnitConversions(astheymayappearonanexamformulasheet) CommonImperial ImperialandMetric MetricLength 1𝑚𝑖𝑙𝑒 = 1760𝑦𝑎𝑟𝑑𝑠

1𝑚𝑖𝑙𝑒 = 5280𝑓𝑒𝑒𝑡1𝑦𝑎𝑟𝑑 = 3𝑓𝑒𝑒𝑡1𝑦𝑎𝑟𝑑 = 36𝑖𝑛𝑐ℎ𝑒𝑠1𝑓𝑜𝑜𝑡 = 12𝑖𝑛𝑐ℎ𝑒𝑠

1𝑚𝑖𝑙𝑒 ≈ 1.609𝑘𝑚1𝑦𝑎𝑟𝑑 ≈ 0.9144𝑚1𝑓𝑜𝑜𝑡 ≈ 0.3048𝑚1𝑓𝑜𝑜𝑡 ≈ 30.48𝑐𝑚1𝑖𝑛𝑐ℎ ≈ 2.54𝑐𝑚

1𝑘𝑚 = 1000𝑚1𝑚 = 100𝑐𝑚1𝑐𝑚 = 10𝑚𝑚

Mass(Weight)

1𝑡𝑜𝑛 = 2000𝑝𝑜𝑢𝑛𝑑𝑠1𝑝𝑜𝑢𝑛𝑑 = 16𝑜𝑢𝑛𝑐𝑒𝑠

1𝑝𝑜𝑢𝑛𝑑 ≈ 0.454𝑘𝑔1𝑜𝑢𝑛𝑐𝑒 ≈ 28.35𝑔

1𝑡 = 1000𝑘𝑔1𝑘𝑔 = 1000𝑔

CommonAbbreviations

𝑚𝑖𝑙𝑒 = 𝑚𝑖𝑦𝑎𝑟𝑑 = 𝑦𝑑𝑡𝑜𝑛 = 𝑡𝑜𝑛𝑓𝑒𝑒𝑡 = 𝑓𝑡𝑜𝑟′𝑖𝑛𝑐ℎ = 𝑖𝑛𝑜𝑟"𝑝𝑜𝑢𝑛𝑑 = 𝑙𝑏𝑜𝑢𝑛𝑐𝑒 = 𝑜𝑧

𝑘𝑖𝑙𝑜𝑚𝑒𝑡𝑟𝑒 = 𝑘𝑚𝑚𝑒𝑡𝑟𝑒 = 𝑚𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑟𝑒 = 𝑐𝑚𝑚𝑖𝑙𝑙𝑖𝑚𝑒𝑡𝑟𝑒 = 𝑚𝑚𝑡𝑜𝑛𝑛𝑒(𝑚𝑒𝑡𝑟𝑖𝑐𝑡𝑜𝑛) = 𝑡𝑔𝑟𝑎𝑚 = 𝑔

Formula

Triangles LinesTrigonometry:

𝑠𝑖𝑛𝐴 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑐𝑜𝑠𝐴 =𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑡𝑎𝑛𝐴 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

PythagoreanTheorem:𝑎2 + 𝑏2 = 𝑐2

Theequationofaline:Slope-interceptform:

𝑦 = 𝑚𝑥 + 𝑏StandardForm:

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0Point-slopeform:

𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)Slopeformula:

𝑚 =𝑦2 − 𝑦1𝑥2 − 𝑥1

2-DimensionalFigure Perimeter AreaRectangle

𝑃 = 2𝑙 + 2𝑤or

𝑃 = 2(𝑙 + 𝑤)

𝐴 = 𝑙𝑤

Triangle

𝑃 = 𝑎 + 𝑏 + 𝑐

𝐴 =𝑏ℎ2

𝐴 =12𝑏ℎ

Circle

𝐶 = 2𝜋𝑟or

𝐶 = 𝜋𝑑

𝐴 = 𝜋𝑟2

3-Dimensionalfigures:RightCylinder:

𝐴FGH = 𝜋𝑟2

𝐴IGFFGJ = 𝜋𝑟2𝐴KLMN = 2𝜋𝑟ℎ

𝑆𝐴FGFPQ = 2𝜋𝑟2 + 2𝜋𝑟ℎ

𝑉 = (𝐵𝑎𝑠𝑒𝐴𝑟𝑒𝑎)ℎ

Sphere:

Sphere:𝑆𝐴 = 4𝜋𝑟2

𝑉 =43𝜋𝑟

Hemisphere:

Hemisphere:𝑆𝐴 = 3𝜋𝑟2

𝑉 =23𝜋𝑟

Side:𝐴 = 𝜋𝑟𝑠Base:𝐴 = 𝜋𝑟2

𝑆𝐴 = 𝜋𝑟2 + 𝜋𝑟𝑠

𝑉 =13(𝜋𝑟2)ℎ

Square-BasedPyramid:

𝐴FWLPXYQN =12𝑏𝑠

𝐴IPKN = 𝑏2

𝑆𝐴 = 2𝑏𝑠 + 𝑏2

𝑉 =13(𝑏𝑎𝑠𝑒𝑎𝑟𝑒𝑎)ℎ

Rectangular(Right)Prism:

𝑆𝐴 = 2(ℎ𝑙 + 𝑙𝑤 + ℎ𝑤)

𝑉 = 𝑙𝑤ℎ

Or𝑉 = (𝑏𝑎𝑠𝑒𝑎𝑟𝑒𝑎)ℎ

TriangularPrism

𝑆𝐴 = 𝑠𝑢𝑚𝑜𝑓𝑎𝑙𝑙𝑓𝑎𝑐𝑒𝑠

𝑉 = (𝑏𝑎𝑠𝑒𝑎𝑟𝑒𝑎)ℎ

MeasurementinTwoSystems.

TheInternationalSystemofUnits(SI)(MetricSystem)

SystèmeInternationald’unités

TheImperialSystemofUnits

History:FormallycalledSystemInternationalebutmorecommonlycalledTHEMETRICSYSTEM.Basedonthemetre.Onemeterisdefinedasthedistancelighttravelsin1/299792458ofasecond.

History:ThesystemusedbytheBritishEmpireandthereforemanyCommonwealthcountriesformanyyears.Tothisdaymuchdailyworkintradesisstilldoneusingtheimperialsystem.Mosttechnicalwork,however,usesthemetricsystem.

Prefixes:Prefixesareaddedtothebaseunitstobeusedwithsmallerorlargermeasurements.teragigamegakilohectodecaBASEUNIT(metre/gram/litre)decicentimillimicronanopico

Some useful conversions:

1 inch = 2.54 cm 1 foot = 30.5 cm (30.48) 1 yard = 3 feet 1 yard = 0.915 m 1 mile = 1760 yards 1 mile = 1.6 km 1 kg = 2.2 lbs 1 litre = 1.06 quarts (US) 1 gallon (US)= 3.79 litres

[1 gallon (UK)=4.55 litres]

TheImperialSystemofUnitsUNIT

QUANTITYMEASURED(circleone)

REPRESENTATIVEEXAMPLE

orREFERENT(acomparisonyoucoulduse)

3EXAMPLESOFOBJECTSYOUWOULDMEASUREUSINGTHISUNIT

MASS?VOLUME?DISTANCE?

1.2.3.

GALLON

Thelargeplasticjugofmilkatthegrocerystore.

1.2.3.

ConvertingBetweenUnitsWithintheImperialSystemConversionFactor:Multiplyingordividingbythisnumberallowsustoconvertfromoneunittoanother.Eg.Convert57inchestofeet.

57𝑖𝑛𝑐ℎ𝑒𝑠 ×1𝑓𝑜𝑜𝑡

12𝑖𝑛𝑐ℎ𝑒𝑠 =5712 𝑓𝑒𝑒𝑡

4 ^_`= 4 U

OneUnitConversionsConvertthefollowing.Answerinexactform(fractionornon-roundeddecimal).

1. 3yd.=___________feet.3yd× 3ft

1yd=9ft

2. 15yd.=___________feet. 3. 12.5yd.=___________feet.

4. 12ft.=___________yards. 5. 2.25ft.=___________inches.2.25ft.× 12in

1ft=27in

6. 136ft.=___________yards.

7. 8ft.=___________inches. 8. 2.75ft.=___________inches. 9. 4.8ft=___________inches.

10. 36in.=___________feet. 11. 140in.=___________feet. 12. 2016in=___________feet.

13. 2mi.=___________yards. 14. 4.2mi.=___________feet. 15. 1500yd.=___________miles.

16. 5250yd.=___________inches. 17. 160oz.=___________pounds. 18. 220oz.=___________pounds.

19. 4lb.=___________ounces. 20. 2.25lb.=___________ounces. 21. 6000lb.=___________tons.

22. Mr.Splaced32yardsticksendtoendacrosshisfrontyard.Findthewidthofhisyardinfeet.

23. Maisycanfit8blocksofbutterinherbackpack.Butterissoldin1poundblocks.HowmanyouncesdoesMaisycarry?

Usethenumbersinthetableonpage5.

1𝑓𝑜𝑜𝑡12𝑖𝑛𝑐ℎ𝑒𝑠

The unit on top is the one you are converting to!

24. AuntieDeeismakingaframeforaphotograph.Theouterdimensionsare3ft.by5ft.Howmanyinchesofframemustshepurchase?

25. Mr.Jwantstoupdatehislivingroomwithcrownmoulding.Theroomisrectangularandmeasures180in.by260in.Mouldingissoldbythefootandcosts$2.19perlinearfoot.Whatisthecostofmouldingrequired(notincludinganytaxes)?

Converteachofthefollowingmeasurementstotheindicatedunits.26. 140feettoyardsandfeet.

Recall: 3yd=1ft140ft×_iM

=1403 𝑦𝑑

= 46 `U𝑦𝑑

140ft=46ydand2ft.

27. 256feettoyardsandfeet. 28. 356inchestoyards,feetandinches

29. 142inchestofeetandinches. 30. 204inchestoyardsandfeet. 31. 84260ouncestotons,poundsandounces.

32. ShelbytheFrenchBullDogneedsanoutdoorareatorun.Benplansonbuildingherthepenbelow.Thefencingmaterialissoldbythelinearfootbuthismeasuringtapeonlyshowsinches.Howmanyfeetwillheneedtopurchase?

33. Convertyouranswerabovetoyards,feetandinches.

34. Whatarethedimensionsoftherectangularcarpetbelowinfeet?

35. Ifthecarpetissoldfor$4.25persquarefoot,whatisthecostofcarpetrequired?Include12%tax.

36. Whatistheperimeterofthegardenshedinyardsandfeet?

37. Theshedcoversone-thirdoftheareaoftheyard.Howmanysquarefeetofsod(grass)areshownintheyardabove?

38. Sod-Warehousesellssodbytheroll.Eachrollis1footwideand4feetlong.Eachrollsellsfor$2.75.whatisthecost(including12%tax)tobuysodfortheyard?

TwoUnitConversions{Becomfortableworkinginfractionformandalwaysreduce.}YouwillneedtouseTWOconversionfactors.Simplyfollowthestepsforoneunitconversions,thenrepeat.Eg.Convert58inchestoyards.�58𝑖𝑛𝑐ℎ𝑒𝑠 × _jGGF

_`LXmnNK= op

_`𝑓𝑒𝑒𝑡

_`𝑓𝑒𝑒𝑡 × _iPWM

UjNNF= op

Uq𝑦𝑎𝑟𝑑𝑠

= 1 ``

Uq𝑦𝑎𝑟𝑑𝑠= 1 __

_p𝑦𝑎𝑟𝑑𝑠

39. 6025feet=_________miles

6025feet× 1yard

3feet= 6025

3yards

60253yards× 1mile

1760yards= qw`o

o`pwmiles

1 7455280

= 1 1491056

𝑚𝑖𝑙𝑒𝑠

40. 123450feet=_________miles

41. 2 12miles=______________inches.

42. 3.25yards=_________inches

43. 15 23yards=_________inches

473yd× 3ft

1yd× 12in

1ft= (ax)(U)(_`)

Uinches

=564inches

44. 24 13yards=_________inches

45. Acabinetmakerisusing1”x3”edgegrainfirtomakesomecabinetdoors.Hepurchased42 2

3yardsonCraig’slist.How

manyinchesdidhebuy?

46. Garyisbuildingpictureframestosellinamarket.Hehas75yardsofmaterialandwillmakesquareframeswithsidelengthsof14inches.Howmanyframescanhemake?

Mr.J’smeasuringwheelclicksonceforeveryyardittravels.Onawalktoschool,thewheelclicks35200times.

47. Howmanyinchesdoeshewalk?

48. Howmanymiles?

49. **Apieceofpaperisfoldedinhalfrepeatedly.Thepaperhasathicknessof1250".Howmanyyards

thickwillthepaperbeafter20folds?

ConversionFactors:

Step� _jGGF_`LXmnNK

Step � _iPWMUjNNF

Note: I combined both steps into one equation here.

ConversionswithNon-linearMeasurements.

50. Howdoyoucalculatetheareaofasquare?

51. Whatistheareaofthesquaretotheleftinsquare-feet?

52. Whatisthesidelengthofthesquareininches?

53. Whatistheareainsquareinches?

54. Toconverttheareaofafigurefromsquarefeettosquareinches,whatcalculationsmustyouperform?

55. Arectangularplotoflandhasdimensionsof0.5milesby0.4miles.Whatistheareainft2?

56. Toconverttheareaofafigurefromsquaremilestosquarefeet,whatcalculationsmustyouperform?

57. Apizzahasanareaof1.5ft2.Ifthepizzaistobeslicedintosixequalslices,howmanysquareinchesiseachslice?

58. Acircularcarpethasanareaof100𝜋ft2(approximately314ft2).Whatisthelengthoftheradiusininches?

59. Thecross-sectionofaconcreteundergroundpipeisshownbelow.Calculatetheareaoftheinner(open)partofthepipetothenearestin2.

60. Thepipeinthepreviousquestionhasconcretewallsthatare5feetthick.Calculatethecross-sectionalareaofconcretetothenearestin2.

TheInternationalSystemofUnits(SI)UNIT QUANTITY

MEASUREDREPRESENTATIVEEXAMPLE 3EXAMPLESOFOBJECTSYOUWOULDMEASURE

USINGTHISUNITMilligram

ONEGRAINOFSALT

Gram MASS?VOLUME?DISTANCE?

ONEPAPERCLIP

Kilogram MASS?VOLUME?DISTANCE?

ONEPINEAPPLE

Millilitre MASS?VOLUME?DISTANCE?

Litre MASS?VOLUME?DISTANCE?

Millimetre MASS?VOLUME?DISTANCE?

Centimetre MASS?VOLUME?DISTANCE?

Metre MASS?VOLUME?DISTANCE?

Kilometre MASS?VOLUME?DISTANCE?

ConvertingBetweenUnitsintheMetricSystem(SI)ConversionFactor:Multiplyingordividingbythisnumberallowsustoconvertfromoneunittoanother.Eg.Convert230mmtocm.

230𝑚𝑚 ×1𝑐𝑚10𝑚𝑚 = 23𝑐𝑚

OneUnitConversionsConvertthefollowing.Answertothenearesttenthwhennecessary.

61. 1250mm=___________cm

1250mm× 1mJ

10JJ=125cm

Or...simplymovethedecimaloneplacetotheleft.

62. 37.25m=___________cm

63. 0.8cm=___________mm.

64. 138m=___________mm 65. 1508cm=___________m 66. 3.28cm=___________mm

67. Acirclehasaradiusof10cm.Findthecircumferenceofthecircleinmillimetres.

68. Afarmerbuildsafencearoundarectangularsheeppen.Thepenis5metreslongand7metreswide.Whatistheperimeterofthepenincentimetres?

69. Findtheheightofatrianglewithabaseof12cmandanareaof75cm2.Answerinmillimetres.

Usethenumbersinthetableonpage5.

1𝑐𝑚10𝑚𝑚

Metric conversions can be made by moving the decimal left or right. Imperial conversions cannot.

Two(orthree)UnitConversionsUsetwoconversionfactorstomakenecessaryconversions.Roundtothenearesttenthofaunitifnecessary.

70. Convert3.45mtomm. 71. Convert12.357kmtom. 72. Convert176mmtom.

73. Convert1.365kmtomm. 74. Convert17 15mtomm. 75. Convert3

4kmtocm.

76. Thepoliovirusisabout30nanometersindiameter.Thatis0.000000030m.Howmanymillimetresindiameteristhevirus.

77. TheGreatWallofMingDynastyinChinahasbeenmeasuredtobe8851.8kmlong.Approximatelyhowmanycentimetresisthis?

78. Astandardvolleyballcourtis18mlongand9mwide.Findtheareainsquaremillimetres.

79. 2m+30cm+4mm =__________________mm

80. 1.35km+125m+40cm =__________________m

81. 1.35km+125m+120mm =__________________cm

UnitConversionbetweenSystemsOne-StepConversions:Usethetableonpage5.Writetheconversionfactorsasaratio(watchtheunits!).Converteachofthefollowing.Roundtothenearesttenth.

82. 50mi=_____________kmUse:1.609}J

1JL(fromreferencepage)

50mi×1.609km1mi =80.45km

≅80.5km

83. 185lb=___________kg 84. 150m=____________yd

85. 72in=_____________cm 86. 42𝑜𝑧 = _____________𝑔 87. 1245km=_____________mi

Two-SteporThree-StepConversions:Converteachofthefollowing.Roundtothenearesttenth.

88. 42km=__________________ft � Convert from kmàmi � Convert from mi à ft

42km×1mi

1.609km×5280ft1mi

≅137824.7ft

89. 54m=____________in 90. 1250g=______________lb

91. Answerthequestionaboveusingadifferentconversionstrategy.

92. 4.25km=_____________in 93. 1.3tons=_____________kg

Convertthefollowing.Exactanswersorroundtothenearesthundredthwhennecessary.

94. 12𝑙𝑏𝑠3𝑜𝑧 = _________𝑘𝑔�Convert3oztolbs.

3𝑜𝑧 ×1𝑙𝑏16𝑜𝑧 = 0.1875𝑙𝑏

�Add:12 + 0.1875 = 12.1875𝑙𝑏�12.1875𝑙𝑏 × w.aoa}Y

_QI= 5.53kg

95. 2𝑙𝑏𝑠14𝑜𝑧 = _________𝑘𝑔 96. 7𝑙𝑏𝑠8𝑜𝑧 = _________𝑔

97. 12’6”=_________m 98. 8yd3’=_________m 99. 14m28cm=__________yd

103. Apizzahasacircumferenceof5feet3inches.Findthediameterincentimetres.

104. Avolleyballhasadiameterof2feet2inches.Findthecircumferenceoftheballatitswidestpoint.Answertothenearestinch.

105. Mr.Jneeds2m,41cmand3mmofedgegrainfirtomakeeachofhiscabinetdoors.Howmanylinearfeetdoesheneedtomakehis8doors?

Geometryof3-Dfigures.Familiarizeyourselfwiththeshapes,namesandformulasatthebeginningofthisbooklet.Usingthereferencepageatthebeginningofthisunit.

• Choosetherightformula.• Fillinallknownvaluesintotheformula.• Calculate(rememberBEDMAS).

SurfaceAreaAreaisthetwo-dimensionalsizeofasurface.Considertheareathatthisbookletiscoveringonthesurfacebelowit(unlessyouareonacomputerofcourse).Thesurfaceareaofasolidisthetotalareaofitsexposedsurfaces.Consideracommoncerealbox.Ifyouunfoldedthesides,top,bottom,frontandback...howmuchareaonyourdeskwoulditcover?Thatwouldbethesurfaceareaofthebox.

127. Refertopage5toanswerthefollowingquestion.

Findthesurfaceareaoftherectangularprismbelowtothenearestsquaremetre.

Findthesurfaceareaofthesquarepyramidbelowtothenearesttensquareinches..

Findthesurfaceareaoftherectangularprismbelowtothenearestsquaremetre.

𝑆𝐴 = 2(ℎ𝑙 + 𝑙𝑤 + ℎ𝑤)𝑆𝐴 = 2[(1.5 × 3) + (3 × 2) + (1.5 × 2)]𝑆𝐴 = 2[4.5 + 6 + 3]𝑆𝐴 = 27𝑚`

Findthesurfaceareaofthesquarepyramidbelowtothenearesttensquareinches..

𝐴 = 2𝑏𝑠 + 𝑏` Need‘s’𝐴 = 2(75)(𝑠) + (75)` Use𝑎` + 𝑏` = 𝑐` 37.52 + 302 = 𝑠2 𝑠 = 48.0𝐴 = 2(75)(48.0) + (75)`𝐴 ≅ 12830𝑠qin

Calculatethesurfaceareaofthefollowingfigures.Answersshouldbegivenasindicated.129. Nearestsquareinch.

130. Nearestsquareinch.

Calculatethesurfaceareaofthefollowingfigures.Answersshouldbegivenasindicated.131. Nearestsquaremillimetre.

132. Nearestsquarecentimetre.

133. Nearestsquarefoot..

134. Nearestsquarefoot.

135. Calculatethesurfaceareaofaconewithaheightof10cmandabasediameterof12cm.Answertothenearestsquarecentimetre.

136. Aconehasabaseradiusof15inchesandasurfaceareaof1650squareinches.Findtheslantheightoftheconetothenearestinch.

137. Frankneedstofindthesurfaceareaoftheplaygroundequipmentbelowsohecanestimatehowmuchpainttobuy.Explaintheprocessheshoulduse(hewillnotpaintthebottom).

Explain________________________________________________________________________________________________________________________________________________________________________________________________________________________________

138. Findthesurfaceareaofthefiguretothelefttothenearestsquareinch.

139. Onequartofpaint(asmallcan)covers87.5sqft.HowmanyquartswillFrankneedtobuy?

140. Findthesurfaceareaoftheconcretecurbbelow(allsurfaces).Answertothenearestsquarecentimetre.

141. Calculatethesurfaceareaofthehexagonalprism(regular)tothenearestsquaremetre.

VolumeVolumeistheamountof3-dimensionalspacethatafigureoccupiesorcontains.Considerthecerealbox.The“amount”ofBeaconBitsthatfitinsidetheboxisitsvolume.Thisisoftenreferedtoasthecapacityofacontainer.

142. Challenge:IfeachBeaconBitisacube1cmby1cmby1cm,howmanyBitswouldfitinthebox?Startbyfindingouthowmanywouldfitinthebottomlayer.

Findthevolumeoftheboxtothenearestcm3.

Findthevolumeoftheconetothenearestcubicinch.

Findthevolumeoftheboxtothenearestcm3.

𝑉 = 𝑙𝑤ℎ

𝑉 = (27)(8)(35)𝑉 = 7560𝑐𝑚U

Findthevolumeoftheconetothenearestcubicinch.

𝑉 =13(𝜋𝑟`)ℎ

𝑉 =𝜋(55)`(75)

𝑉 =226875𝜋

3 = 75625𝜋 ≅ 237583𝑠𝑞𝑖𝑛

145. Usingtheformulaforthevolumeofarectangularprism,𝑉 = 𝑙𝑤ℎ,explainwhytheunitsforvolumeare“cubed”.

146. Refertopage5.Comparetheformulastocalculatevolumeofrightprisms,rightcylinders,andtriangularprisms.Whatdotheyallhaveincommon?

147. Timothyisbuildingagardenwaterfountainintheshapeofa“truncated”squarepyramid.Describehowyouwouldfindthevolumeofconcreterequiredtobuildthisfountain.

148. Findthevolumetothenearesttenthofmetre.

Calculatethevolumeofthefollowingfigures.Answersshouldbegivenasindicated.149. Cubiccentimetres.

150. Cubiccentimetres.

151. Cubicfeet.

152. Cubicmetres.

153. Nearesttenthofacubicmetre.

154. Aspraynozzleproducesacone-shapedspraypattern.Findthevolumeoftheconeifthenozzleis12feetabovethegroundandproducesacirclewithadiameterof16feet.(Nearestcubicfoot)

155. Findthevolumetothenearestcubicinch.

156. Ifthepyramidtotheleftisenlargedbyafactorof2,whatwillthenewvolumebe?

Canyoumakearule(formula)forthis?

157. Asquare-basedpyramidhasavolumeof250cubicyardsandaheightof30feet.Findthesidelengthofthesquarebasetothenearestfoot.

158. Aspherehasavolumeof3000m3.Findtheradiusofthespheretothenearestmetre.

159. Charlieneedstopaintthecompositeshapebelow.Beforehepurchasespaintheneedstocalculatethesurfaceareatothenearestsquarefoot.Thebottomdoesnotneedtobepainted.Howmanysquarefeetdoesheneedtopaint?

160. Findthevolumeofgravelinthepiletothenearestcubicyard.

161. Findasphericalobjectandmeasurethediameter.Calculatethesurfaceareaofyourobject.Drawaneatanddetaileddiagramshowingyourobjectandmeasurement.Recordallmeasurementstothenearesttenthofacentimetre.Roundyouranswertothenearestsquarecentimetre.

162. Findaconicalobject.Makeappropriatemeasurementstothenearesttenthofacentimetre.Drawaneatanddetaileddiagramshowingyourobjectandmeasurements.Calculatethevolumeoftheconetothenearestcubiccentimetre.

163. Onegallonofpaintcoversapproximately350sq-ft.Howmanydecorativeballscanyoupaintwitha5-gallonbucketofpaintifeachballhasaradiusof12cm.

164. Acylindricalcanholds3tennisballs.Thediameterofatennisballis2½inches.Calculatethevolumeofairinthecansurroundingthe3balls.Thecanisdesignedtoholdexactlythreetennisballsintermsofheightanddiameter.(Nearesttenthofacubicinch).

165. Asectionofconcretepipehasaninsidediameterof2.2mandanoutsidediameterof2.5m.Findthecross-sectionalareaofexposedconcreteforoneendofthepipe.

166. Aspherehasasurfaceareaof260𝜋squarefeet.Findtheexactradiusofthesphere.

167. Aspherehasasurfaceareaof289𝜋squareinches.Findtheradiusofthespheretothenearesttenthofaninch.

168. Belowisamodelofastandardroomatastoragefacility.Theinteriorwallsaretobepainted.Theroommeasures18’by24’andthewallheightis8’.Thedoorisstandardheightandwidth(36”by80”).Findtheinteriorsurfaceareaofthewalls.

Checkoutthehugeaquarium!Mr.Jwantstoswimthefishessoheisbuildingthisaquariuminhishome.

169. Calculatethevolumeofconcreteinthefloorofthe

aquariumincubicfeet.

170. Calculatethemassofthefloorifconcretehasamassof2400kgpercubicmetre.

171. Calculatetheareaofacrylic(aquariumwall)requiredtobuildthisstructure.

172. Sheetsofacrylicsellfor$45/sqft.Ifhecouldactuallyfindsheetsthissize,whatwouldbethecost(beforetaxes)?

173. Howmanylitresofwaterdoesthetankhold?Howmanygallons?Note:1𝑐𝑚U = 1𝑚𝑙, 1000𝑚𝑙 = 1𝑙𝑖𝑡𝑟𝑒,1𝑔𝑎𝑙𝑙𝑜𝑛 = 3.785𝑙𝑖𝑡𝑟𝑒𝑠

174. Aspherehasavolumeof256�3cm3.Findtheexactradius

ofthesphere.

175. Asquare-basedpyramidwithaheightof10metreshasavolumeof300m3.Findtheexactsidelengthofthebase.

176. Findtheheightofacylinderifithasavolumeof1200cm3andaradiusof12cm.Answertothenearesttenthofacentimetre.

177. Findtheexactvolumeoftherightconebelow.Ithasaheightof10cmandaradiusof5cm.

178. Findtheexactvolumeofacylinderwiththesameheightandradiusastheconeinthepreviousquestion.

179. Whatistheratioofvolumesforthetwofiguresintheprevioustwoquestions?

180. Findthevolumeoftherightprismbelow.Answertothenearestcubicinch.

181. Belowisarightpyramidwiththesamebaseandheightastheprisminthepreviousquestion.Findthevolumetothenearestcubicinch.

182. Whatistheratioofvolumesforthetwopreviousfigures?

183. Ingeneral,whatistherelationshipbetweenthevolumesofrightpyramidsandrightprismswiththesamedimensions?

Answers:1. 9ft2. 45ft3. 37 _

`or37.5ft

4. 4yd5. 27inches6. 45 _

7. 96𝑖𝑛8. 33in9. 57.6in10. 3ft11. 11 `

U𝑓𝑡

12. 168ft13. 3520yd14. 22176ft15. xo

ppmiles

16. 189000in17. 10lbs18. 13 U

aor13.75lbs

19. 64oz20. 36oz21. 3tons22. 96’23. 128oz24. 192”25. $162.06for74’26. 46yd,2ft27. 85yd,1ft28. 9yd,2ft,8in29. 11’10“30. 5yd,2ft31. 2tons,1266lbsand4oz32. 74ft33. 24yd,2ft34. 6’x7’35. $199.9236. 13yd,1ft37. 192sqft38. $147.84for48rolls39. 1 149

1056𝑚𝑖𝑙𝑒𝑠(about1.141

miles)40. 23 67

176≅ 23.4𝑚𝑖𝑙𝑒𝑠

41. 158400”42. 117”43. 564”44. 876”45. 1536”46. 48frames47. 1267200”48. 20miles49. 116.5yd50. 𝐴 = 𝑙𝑤51. 9sqft52. 36in53. 1296sqin

54. Multiplybytheconversionfactor(12)twice.Thatis,multiplyby122

55. 5575680sqft56. Multiplyby(5280)257. 36sqin58. 120”59. 101788sqin60. 79168sqin61. 125cm62. 3725cm63. 8mm64. 138000mm65. 15.1m66. 32.8mm67. 628mm68. 2400cm69. 125mm70. 3450mm71. 12357m72. 0.2m73. 1365000mm74. 17200mm75. 75000cm76. 0.00003mm77. 885180000cm78. 162000000mm279. 2304mm80. 1475.4m81. 147512cm82. 80.5km83. 84.0kg84. 164.0yd85. 182.9cm86. 1190.7g87. 773.8miles88. 137824.7ft89. 2126.0in90. 2.8lbs91. 137795.3ft92. 167322.8in93. 1180.4kg94. 5.53kg95. 1.31kg96. 3402g97. 3.81m98. 8.23m99. 15.62yd100. 3.81m101. Tryyards

àfeetàmetres102. Trymàcm

àinchesàyards103. 50.94cm104. 82”105. 66 _

U=66.33linearft.

106. Youranswershere.107. Youranswershere.108. Youranswershere.109. Youranswershere.110. Trundlewheel.111. Verniercalipers.112. Micrometer.113. Inches,cm,mm.114. Eg.Measuringtape.Cm,

inches(andfractionsthereof)

115. Eg.Measuringcup.Cups,ounces,ml,l

116. Eg.Volumetriccylinder.ml

117. Eg.2-panbalancescale.g,kg

118. Eg.Electronicscale.𝜇𝑔,mg,g

119. Diameter,distance.120. Volume.ml121. Diameter,distance.122. Onesetofclawsisfor

measuringinsidediameterssuchasinsideatube.Theotherisformeasuringoutsidediameter.

123. 1.97mm124. 3.15cm125. 3.68cm126. 0.40mm127. 27m2128. 12830sqin129. 81268sqin130. 19684sqin131. 1810mm2132. 86cm2133. 16965sqft134. 137sqft135. 333cm2136. 20”137. Findthesurfaceareaof

therectangularprism,addtheareaofthecurvedcylindricalsurface,subtractthebottomoftheprismandthetwocircles.

138. 41734sqin139. Frankshouldbuy4

quarts.140. 739655cm2(with

bottom)478889cm2(withoutbottom)

141. 36m2142. 7560BeaconBits

143. 7560cm3144. 237583cubicinches145. Thethreedimensions

haveunitssuchascentimetres.Ifwemultiply𝑐𝑚 × 𝑐𝑚 × 𝑐𝑚weget𝑐𝑚U.

146. Bottom(basearea)multipliedbytheheight.

147. Findthevolumeofthelargepyramidandsubtractthevolumeofthesmall(removed)pyramid.

148. 11.5m3149. 11058432cm3150. 4193205cm3151. 85cubicfeet152. 24.4m3153. 7.1m3154. 804cubicfeet155. 60cubicinches156. 480cubicinches157. 26ft158. 9m159. 269sqft160. 4yd3161. Youranswerhere.162. Youranswerhere.163. 897balls164. 12.3cubicinches165. 1.1m2166. Radiusis√65ft167. 8.5in168. 652sqft169. 448cubicfeetor12.69

m3170. 30446kg171. 616sqft172. $27720173. 76116l,20108gal174. 4cm175. 3√10m(approx9.49m)176. 2.7cm177. `ow

U𝜋cm3

178. 250𝜋cm3179. 1: 3or_

180. 212628cubicinches181. 70876cubicinches182. 1: 3or_

183. Pyramidsandconeswillhavevolumesequaltoone-thirdoftheircorrespondingprism.

Measurement in Two Systems · 1.6 Describe and explain a personal strategy used to determine a...

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