Volume Of A Prism - Mr. Ma's Website

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Volume of a PrismFocus on After this lesson, youwill be able to φ use a formula toφdetermine thevolume of a rightrectangular prismuse a formula todetermine thevolume of a righttriangular prismRecycling bins are located in most schools. Students and staff membersfill them with paper, aluminum cans, plastic and glass bottles, and otherrecyclables. Schools probably recycle more paper than anything else.What would you need to know about a bin for recycled paper in orderto determine how much paper was recycled?How can you use the dimensions to calculate the volumesof right prisms?1.How is volume related to the area of the base and the heightof a prism?2.Show how would you could determine thevolume of the right rectangular prism shown?4 cm9 cm5 cm3.Show how you could determine thevolume of the right triangular prismshown?5 cm4 cm8 cm254MHR Chapter 707 ML8 Chapter07 10th.indd 2544/9/08 4:26:36 PM

Reflect on Your FindingsIf you know only the dimensions of a right rectangular prism, howcan you determine the volume? Test your method using an example.b) If you know only the dimensions of the triangular base and theheight of the prism, how can you determine the volume of a righttriangular prism? Test your method using an example.c) How are the formulas for the volume of a right rectangular prismand a right triangular prism different? How are they the same?4. a)Example 1: Use a Formula to Determine the Volume of a RightRectangular Prisma)Determine the volume of theright rectangular prism.b)h 4 cmDetermine the volume ofthe cube.5 cml 2 cm5 cmw 3 cmHow do you calculatethe area of a rectangle?5 cmSolutiona) Volume of a right rectangular prism area of rectangular base height of prismVolume of a right rectangular prism (length width) heightStrategiesV l w hUse a VariableV 2 3 4V 24The volume of the right rectangular prism is 24 cm3.b)A cube is also a right rectangular prism.Volume of a cube area of square base height of prismVolume of a cube (length width) heightV s s sV 5 5 5V 125The volume of the cube is 125 cm3.a)What is the volume of theright rectangular prism?h 5 cmw 4 cmb)Since s s s s3, you couldwrite the formula as:V s3V 53V 125What is the volume of a cubewith edge length 3 m?3ml 8.5 cm3m3m7.2 Volume of a Prism MHR07 ML8 Chapter07 10th.indd 2552554/9/08 4:26:39 PM

Example 2: Use a Formula to Determine theVolume of a Right Triangular Prism2.6 mDetermine the volume of the right triangular prism.StrategiesUse a VariableHow do you calculatethe area of a triangle?SolutionVolume of a triangular prism area of triangular base height of prismVolume of a triangular prism (base height 2) height of prismV (3 2.6 2) 9V 35.1The volume of the right triangular prism is 35.1 m3.What is the volume of the right triangular prism?9m3m5 mm8 mm9 mm30 cmOatCRUNCHIESOatWhen Katie opened a new box of OatCrunchies, she noticed that the box5 full.was only6a) How much space was empty?b) Why do you think packages often seemto have empty space when first opened?CRUNCHIESCRUNCHIESExample 3: Use Volume to Solve a Problem8 cm18 cmSolutiona) The cereal box is a right rectangular prism.V l w hV 18 8 30V 4320The volume of the cereal box is 4320 cm3.256MHR Chapter 707 ML8 Chapter07 10th.indd 2564/9/08 4:26:40 PM

5 full. Therefore,1 of the package is empty.The package is661Amount of empty space 4320665 1– 43206 6 6 6 720There was 720 cm3 of empty space.b) Packages often seem to have empty space because the contents settlewhen being shipped.Mr. Chin bought a box of small building blocks for his four children.He will give an equal number to each of them. What volume of blockswill each child get? The volume of a right rectangular prism can be determinedusing the formula:V l w hl The volume of a cube can be determined using the formula:V s s sV s3 The volume of a right triangular prism can be determinedusing the formula:V (base of triangle height of triangle 2) heightof prismhwsssheightof triangleheight of prismbaseof triangle7.2 Volume of a Prism MHR07 ML8 Chapter07 10th.indd 2572574/9/08 4:26:40 PM

1.Grace tells Dakota that the volume of a cube can be found usingthe formula: V l w h. Do you agree? Explain.2.Kwan wants to build a concrete ramp to his back door. He wantsto determine the volume of concrete needed for the ramp. Whatmeasurements does he need to know? Justify your response.3.Jack’s family opened a full carton of frozen yogurt for dessert.3 left. Jack wants to know what volumeAfter they ate, there was4of frozen yogurt they ate. He does the following calculation.Volume of carton:Volume of frozen yogurt eaten:3V 12 X 9 X 18V 1944 X4V 19441944 X 3The volume of the carton is 1944 cm3.frozen yogurtV 418 cm chocolate5832V 4V 145812 cmThey ate 1458 cm3 of frozen yogurt.9 cmWhat mistake did Jack make?b) Show the correct calculation.c) Show an alternative way to calculate the answer.a)c)For help with #4 to #6, refer to Example 1 on page 255.4.Determine the volume of each rightrectangular prism.a)5m3mb)12 m6m11 m2584m5.1m6.2 m2mWhat is the volume of each rightrectangular prism?a) l 2 m, w 2 m, h 10 mb) l 8 cm, w 7 cm, h 9 cmc) l 11.7 mm, w 6.3 mm,h 2.9 mmd) l 6.2 cm, w 6.4 cm, h 6.4 cmMHR Chapter 707 ML8 Chapter07 10th.indd 2584/9/08 4:26:42 PM

6.Determine the volume of each cube.a)For help with #9 and #10, refer to Example 3 onpages 256–257.b)3 cm9.10 cmDetermine the volume of the contentsof each right prism.31a) fullb) full383 cmc)12 cm2.5 cm19 cm5 cm7.3c)Determine the volume of each righttriangular prism.a)4140 mmfull2.2 mb)7 cm15 cm10 cmFor help with #7 and #8, refer to Example 2 onpage 256.400 mm12 cm260 mm15 m10.7 cm8.1 mc)Determine the volume of the emptyspace in each object.4a) full of facial tissues510 cm10 cm400 mm3b)320 mm8.4full of milk210 mmWhat is the volume of each righttriangular prism?a) base of triangle 3 mheight of triangle 7 mheight of prism 8 mb) base of triangle 15 cmheight of triangle 8 cmheight of prism 20 cmc) base of triangle 10 mmheight of triangle 9.1 mmheight of prism 11.3 mm12 cmMILK2%4 cm1c)6KILM2%7 cmfull of water12 cm24 cm15 cm7.2 Volume of a Prism MHR07 ML8 Chapter07 10th.indd 25910 cm2594/9/08 4:26:42 PM

11.a)Width(cm)7Height(cm)2b) 1215c)Volume(cm3)Height ofTriangle (cm)a) 7Cindy’s aquarium stands 75 cm tall andhas a base that measures 1.2 m 80 cm.At one point during the initial filling, theaquarium has a 12-cm depth of water init. Cindy needs to fill it to 15 cm from thetop before she adds the fish. Draw adiagram and label the dimensions of theaquarium. Determine how much morewater Cindy must add before she puts inthe fish.17.A contractor is excavating a rectangularhole 10 m 12 m 3 m to pour thefoundation for a house. A dump truckwith a capacity of 9 m3 is used to haulaway the excavated soil. How many tripsdoes the truck need to make?7010108051200Height ofPrism(cm)Volume(cm3)10108057002b) 1814c)70A landscaper has 0.5 m3 of gravel to use asthe base of a patio. If the gravel base mustbe 10 cm deep and the patio is 2.6 m wideand 2.8 m long, does she have enoughgravel? How much extra gravel does shehave, or how much more will she need?The word deep is sometimes usedto describe the height.14.16.Copy and complete the following table.Right Triangular PrismBase(cm)13.Calculate the volume of concrete usedto make a sidewalk 1.5 m wide and120 m long. The concrete is poured0.1 m thick.Copy and complete the following table.Right Rectangular PrismLength(cm)12.15.A glass vase in the shape ofa right triangular prism isfilled with coloured sandas a decoration in a livingroom. What is the volumeof the vase?8 cmLiteracy80 cmLinkCapacity refers to the greatest volume that acontainer such as a tank, a truck, or a measuringcup can contain.15 cm260MHR Chapter 707 ML8 Chapter07 10th.indd 2604/9/08 4:26:43 PM

18.Suki has 30 small linking cubes.a) She wants to use 18 of them to makea large cube. Is this possible? Why orwhy not?b) What number of linking cubes wouldshe use to construct the largest cubeshe can possibly make?21.Sketch and label all possible rectangularprisms with dimensions that are wholenumbers of centimetres that have a volumeof 120 cm3. Do not consider a change oforientation as a new figure; for example,4 cm 5 cm 6 cm is the same as6 cm 4 cm 5 cm.19.Melissa has three glass vases. She wantsto use one as a decorative fish tank forHarvey the guppy. Which will give Harveythe most water to swim in?22.A tank made of concrete has outsidedimensions measuring 6 m 3 m 1 m.It has no lid. The concrete is 8 cm thick.What is the maximum volume the concretetank can hold? What assumptions didyou make?23.Rectangular Prism A and RectangularPrism B have the same length. The widthof A is half the width of B. The height ofA is twice the height of B. What is thedifference in volume?24.A rectangular tank, 40 m long by 30 mwide, is filled with 960 m3 of water.a) Determine the depth of water.b) If the water drains out at a rate of60 m3/h, how much water is left after2.5 h? What is the new depth of water?c) Later, the depth of the water is 0.2 m. Forhow long has the tank been draining?7 cm5 cm7 cm7 cm4 cm21 cm10 cm7 cm9 cm20.The ratio of length : width : height of a boxis 6 : 3 : 2. What is its volume if the heightis 5 cm?MATH LINKThe Parks Committee is considering putting 12 of theserecycling bins throughout the park. If the bins are filled tothe brim and emptied twice weekly, what volume of wasteis recycled each week?7.2 Volume of a Prism MHR07 ML8 Chapter07 10th.indd 2612614/9/08 4:26:46 PM

Solution a) Volume of a right rectangular prism area of rectangular base height of prism Volume of a right rectangular prism (length width) height V l w h V 2 3 4 V 24 The volume of the right rectangular prism is 24 cm3. b) A cube is also a right rectangular prism. Volume o

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