Ch2 Handouts

Lesson 2Name: _______________________Ch2.2 Volume and Surface AreaNotes:In general:Volume of any Prism = area of the base x heightSurface area of any Prism = the sum of the area of all sides (surfaces)Volume and Surface Area FormulasGeometric figure Volume Surface AreaRectangular PrismSA = 2lw + 2wh + 2lhV = lwhorSA = 2(lw + wh + lh)Triangular PrismV = 1 2 blhSA = bl + ah + bh + chCylinderSA = 2πr 2 + 2πrhV = πr 2 horSA = 2πr(r + h)

Example 3:Determine the volume, to two decimal places, or concrete needed to construct this staircase.0.9 m 4.5 cmExample 4:A riser is a raised platform used on a stage. This riser, for a rock performance, is to be painted.Determine the surface area to be painted. Do not include the bottom of the riser.12 m10 m6 m16 mHow many faces are there? (do not include the bottom) _________Calculate the area of each face, and then determine the total surface area.

Lesson 3Name: _______________________Ch2.3 Optimizing Areas and PerimetersSuppose a friend asks you to help build a deck for their parents’ cottage. How can you determinethe most suitable size and shape for the deck?Often, there are constraints that must be considered, such as:* the budget* fixed perimeter* fixed area* boundariesAn expert in construction can apply measurement and geometry concepts to optimize the design ofa deck or other structures.Investigation A: Maximizing Area given Perimeter (4-sided)Suppose you have twenty-four 1-m sections of edging to build a rectangular deck. What is themaximum area that you can enclose?DeckEach toothpick represents a section of edging. Construct as many rectangles as you can, using all24 toothpicks. For each rectangle, record the dimensions, area, and perimeter in a table like this:Perimeter (m) Length (m) Width (m) Area (m 2 )24 10 2 202424242424What are the dimensions of the deck with the maximum area? ___________________What is the maximum area? _______________What type of rectangle will provide the maximum area for a given perimeter? ______________To maximize the area (given a fixed perimeter) for a 4-sided rectangular structure,the dimensions are calculated as followsFORMULA:Width = P 4Length = P 4

What if you were only enclosing three sides rather than four?Investigation B: Maximizing Area given Perimeter (3-sided)Ryan has a summer job working at a campground. He is roping off a rectangular swimming areaon the beach at Swan Lake, as shownSwan lake beachIn the equipment shed, Ryan found a 100 m rope.He does not need to rope along the beach, so he onlyNeeds to make 3 sides of a rectangle in the water.Swimming areaHint: there will be one length and two widths,Perimeter (m) Length (m) Width (m) Area (m 2 )100 80 10 800100100100100100What are the dimensions of the maximum swimming area? ___________________What is the maximum possible swimming area? _______________What type of rectangle will provide the maximum area for a given perimeter? ______________To maximize the area (given a fixed perimeter) for a 3-sided rectangular structure,the dimensions are calculated as follows:FORMULA:Width = P 4Length = P 2

What if we were given a fixed area, rather than a fixed perimeter?Let’s investigate how we would minimize perimeter for a given area.Investigation C: Minimizing Perimeter given Area (4-sided)Latoya wants her new vegetable garden to have an area of 64 m 2 . The garden will be rectangularand surrounded by a fence. What dimensions will require the least amount of fencing, to thenearest hundredth of a metre?We now have a fixed area, and are looking for the dimensionsthat minimize the perimeter.GardenArea (m 2 ) Length (m) Width (m) Perimeter (m)64 16 4 4064646464What are the dimensions for the least amount of fencing? ___________________What is the minimum perimeter for the garden? _______________What type of rectangle will provide the minimum perimeter for a given area? ______________To minimize the perimeter (given a fixed area) for a 4-sided rectangular structure,the dimensions are calculated as follows:FORMULA:Width= ALength= AYou may not always get a whole number when you square root the given area.In that case, round to two decimal places.Note:___________________________________________________________________________If we need to minimizing perimeter given area for 3-sided rectangular structures, we can use thefollowing formula:Width =A2ALength = 22

Lesson 4Name: _______________________Ch2.6 Optimizing Volume and Surface AreaYesterday you learned how to maximize area and minimize perimeter of a rectangle.Today, you will learn how to minimize surface area and maximize volume of 3-D shapes.1) SQUARE-BASED PRISMA) Minimizing Surface area with a given Volume:Summary:For a square-based prism with a given volume, the minimum surface area occurs whenthe prism is a cube.Steps: 1) sidelengths 3V22) SA 6sminExample 13A square-based prism has a volume of 8000m , what dimensions will produce the smallest surfacearea? What is the smallest (minimum) surface area?B) Maximizing Volume with a given Surface Area:Summary:For a square-based prism with a given surface area, the maximum volume occurs whenthe prism is a cube.Steps: 1)2)sidelengths Vmax s3SA6Example 22A square-based prism has a surface area of 150cmvolume? What is the largest volume?, what dimensions will produce the largest

2) CYLINDERA) Minimizing Surface area with a given Volume:Summary:The maximum volume for a given surface area of a cylinder occurs when its height equals itsdiameter: h = d or h = 2r (replace h with 2r in the V formula, and simplify)VSteps: 1) radius r 322)SAmin 6r2Example 13The volume of a cylinder is 1356.48cm, what dimensions will produce the smallest surface area?What is the smallest surface area?B) Maximizing Volume with a given Surface Area:Summary:The minimum surface area for a given volume of a cylinder occurs when its height equalsits diameter: h = d or h = 2r (replace h with 2r in the SA formula, and simplify)SASteps: 1) radius r 262) Vmax 2r3Example 22The surface area of a cylinder is 75.36m, what dimensions will produce the largest volume?What is the largest volume?