I Practise. O At a lake, a rectangular swimming area is to be roped in on. Key Concepts. Communicate Your Understanding

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1 less, or the same amount of area as before? Investigate A. Will the hedge allow Brandon to enclose more, c) Compare this result with the maximum area you found in k. 9. Perimeter and Area Relationships of a RectangleS MHR 87 c) 5Dm a) Gm b) 6 m d) 8 m with each perimeter? 1. What dimensions will provide the maximum area for a rectangle I Practise a a a a a i 5 a a S length and width of this optimal area be related? three sides to create the greatest area possible. How will the b) When does a square dot maximize the enclosed area? a) When does a square maximize the enclosed area? that will allow the farmer to maximize the enclosed area. A farmer wants to fence a rectangular field. Suggest two things O At a lake, a rectangular swimming area is to be roped in on Communicate Your Understanding a greater area can be enclosed. Optimizing the area of a rectangle means finding the dimensions number of sides to be fenced. If fencing is not required on all sides, The dimensions of a rectangle with optimal area depend on the result in the maximum area. For a rectangle with a given perimeter, there are dimensions that of the rectangle with maximum area for a given perimeter. Key Concepts of three sides? 6. Reflect How can you predict the dimensions of a rectangle with maximum area if you know the sum of the lengths your hypothesis? b) Test your prediction. Were you correct? Do you need to change a) Predict the dimensions of the rectangle with maximum area. 5. Suppose Brandon has 0 m of prefabricated fencing to work with. that you found. Describe any relationship that you notice.. Examine the length and width of the enclosure of maximum area

2 For the fence materials in each of parts a) and b) how much c in length. What is the maximum rectangular area that you can Draw a diagram and label the dimensions of the maximum Talia can rope off. enclose with there is an existing wall that will be used as one of the boundaries.. A farmer is adding a rectangular corral to the side of 5. A fence is to be built with prefabricated sections that are.8 m 6. A fence is being built using the materials in question 5, but now b) 0 pieces? spreadsheets, The Geometer s Sketchpad, or a graphing calculator. a barn. The barn will form one side of the rectangle. The farmer of the barriers? c) How much more area can be enclosed if the rope is used instead same area be enclosed? Explain. the dimensions of the corral with maximum area. Use any tools: a) What are the dimensions of the rectangle of maximum area? light to enter the room? has 16 m of fencing to use. Conduct an investigation to determine b) What window dimensions will allow the maximum amount of b) Suppose 1 barriers, each m long, are used instead. Can the a) Sketch three different windows that have a perimeter of 6.0 m. a) 0 pieces? must be 6.0 m. Include dimensions. toothpicks, geoboards, grid paper, tables, or technology such as told her to section off a rectangular area in a corner of the warehouse. Conduct an investigation to determine the greatest area that company. Her uncle gave Talia 0 m of rope and warehouse and he has given Talia an area in which to store (lie computer supplies for his 7. Chapter Problem Talia s uncle owns a rectangular area that you can enclose with additional area does using an existing border provide? To keep the cost as low as possible, the perimeter of the window. A rectangular enclosure is to be created using 8 m of rope. b) 0 pieces a. To brighter a room, a rectangular window will be built into a wall. a) 0 pieces Cnnnnnlln Reflecting -01 I Connect and Apply r t 88 MHR Chapter 9 N Preblem Salving Representing Selecting reels Rcas,un and Prcvlng

3 fourth side four sides three sides, using a hedge at the back of the property as the 9. Perimeter and Area Re ationships of a RectangleS MHR 89 b) the maximum area of a rectangle for a given perimeter a) the minimum perimeter of a rectangle for a given area 11. Describe a situation in which it is important to know solve it. the area of a garden. Solve the problem and then have a classmate 10, Pose a probkm involving the relationship between the perimeter and the playground? the playground? c) What is the minimum length of fence that can be used to enclose b) What dimensions use the minimum length of fence to enclose =8+1 :7/8 7 =0 + 8 I I J 7/8 7 =C+*8 + 1 I Rectangle Width (m) Length Cm) Area (m) Length of Fence Used (m) I:{i LEB1QI pthfliiqi a) Investigate other possible rectangles with an will form one side of the rectangle. The area of playground to the side of a school. The school 9. A conlxactor is adding a rectangular kindergarten and calculate the maximum area that can be enclosed in each case. rectangle is shown. Draw diagrams for each of the three scenarios in Brandon s proposal the playground is to be 7 in. One possible area of 7 m. Copy and complete the table on an adjacent side or use a spreadsheet like the one shown. Sm two sides, using the hedge and an existing neighbour s fence available can be used to fence an enclosure on In the proposal, lie reports how the m of fencing that is 8. Brandon prepares a proposal for his client. H

4 Extend perimeter. and want to determine the maximum area that can be enclosed by three adjoining rectangular fields that have the greatest 0ssible area. Determine the dimensions of the three fields are the dimensions of the maximum c) your results in a table. What are the area that can be enclosed. Record enclose one side. Use diagrams or toothpicks to determine the maximum Suppose an existing hedge is used to dimensions of this shape? Suppose two perpendicular hedges shape? What are the dimensions of tins optimal enclosed if fencing is used on all four sides? enclose the area on two sides. What Conduct an investigation to determine the shape of the yard with diameter 0 cm. with the string. A rectangular yard with an area of 50 m is to be fencçd on three Is Ranjeet correct? If so, find the maximum area that can be enclo5 is a square. So, the maximum area for a square with sides 6 cm each is 6 cm. Ranjeet replied, I can make a figure with a greater area. 15. If a triangle is drawn inside a circle so that the three vertices touch 16. Math Contest Find the dimensions of the rectangle of maximum minimum perimeter sides. Minimizing the perimeter will minimize the cost of the fence. triangle of maximum area that can be inscribed in a circle with but do not intersect the circle boundary, then the triangle is inscribed in the circle. Conduct an investigation to find the dimensions of the rectangular floor of a toolshed with area 5 m and minimum Conduct an investigation to determine the dimensions of the a) What is the maximum area that can be area that can be enclosed? area that can be inscribed in a circle of radius 10 cm. Math Contest Katrina and Ranjeet have a piece of string cm long the string. Katrina said, The shape that will give the maximum area of fencing. A rectangular area is to be enclosed with 1 m 18. Math Contest A farmer has 500 m of fencing. He wants to constrr]t b) 1. * Achievement Check lii 90 MHR Chapter 9 10 mz, building ith area greater than permit is required for any new Act of Ontario, a building According to the Building Code m Did You Know?

5 where a cube-shaped box is not the most desirable shape. why are all boxes not cube-shaped? Give an example of a situation O If a cube-shaped box requires the least amount of material to make, area for a given volume. 9. Minimize the Surface Area of a Square-Based PrismS MI-fR 95. Determine the dimensions of the square-based prism box with each. Determine the surface area of each prism you found in question, to the nearest square centimetre to the nearest tenth of a centimetre. For help with question, see Example. Determine the dimensions of a square-based prism container with volume 00 cm and minimum heat loss. Round the dimensions. a) 51 cm b) 1000 cm c) 750 cm d) 100 cm dimensions to the nearest tenth of a centimetre, when necessary. volume that requires the least material to make. Round Ihe For help ti ith questions and, see Example 1. II II II Ii II II I II Camnunlailne Conooctin 1 Ref Incline Box A Box B Box C I So1a Explain your reasoning. Pop 000flc!o Sc ILaJflO Tools -N Rank them in order from least to greatest surface area. RasonInc and Pinalne 1. These square-based prisms all have the same volume. Practise a Describe a situation when you would need to minimize the surface Communicate Your Understanding

6 - ji ] ] ) Connect and Apply 96 MI-JR Chapter 9 iii mind. for its juice boxes, keeping your results from parts a) and b) c) Does the lidless box require more, less, or the same amount c) Write a letter to the manufacturer recommending a new design with the dimensions you found in part a). b) Compare your results to those in question 7. Are b) Suggest reasons why juice boxes are not usually manufactured hint: 1 ml = 1 cm. of material to construct, compared to the box with a lid? Round the dimensions to the nearest tenth of a centimetre. holds 00 nil of juice and requires Ge least amount of material. 9. a) Determine the dimensions of a square-based prism juice box [hat the dimensions the same or different? j[j F surface area and a capacity of.5 L. dimensions of a lidless box with minimum Popcarn a) Carry out an investigation to determine the 8. Reo queion 7. Urnlly, when you buy popcorn [Tint: I L = 1000 cm. to be a square-based prism with a capacity of.5 L. 7. A movie theatre wants its large box of popcorn will need, to the nearest tenth of a square centimetre? Determine the least amount of cardboard required to b) What is the minimum amouni of cardboard that Tuba a) What should the dimensions of the box be, to the Talia wants to use the minimum amount of cardboard when she ships the box. box must have a capacity of 750 cm and cardboard squared-based prism box. The construct this box, to [lie nearest square centimetre. (universal serial bus) cables in a small nearest hundredth of a centimetre? 6. Chapter Problem Talia is shipping USH [U U1* for this. one you found in part a)? Suggest reasons b) Does laundry detergent usually come in a box shaped like the dimensions to the nearest tenth 01 a centimetre. the box require [lie least amount of cardboard? Round [lie a) The box contains 000 cm of detergent. What dimensions for 5. Laundry detergent is packaged in a square-based prism box.

7 . The maximum volume for a given surface area of a square-based The surface area of a cube is given by the formula SA = 6s, where s prism always occurs when the prism is a cube. Key Concepts 9. Maximize the Volume of a Square-Based Prism MHR 501 Box A Box B Box C prisms in order of volume from greatest to least. 1. The three square-based prisms have the same surface area. Rank the Practise L... BoxA Boxu BoxC the greatest volume? Explain how you know. 0 0 maximum volume of a square-based prism, given its Describe a situation where it would be necessary to surface area. These three boxes all have the same surface area, Which box has find the Communicate Your Understanding maximum volume solve for s to find the dimensions of the square-based prism with is the side length of the cube. When you are given the surface area, and a height exist that result in the maximum volume I For a square-based prism with a given surface area, a base length

8 a) 150 cm tenth of a unit when necessary. For lie/p with questions and, see the Example. 50 MHR Chapter 9 to the nearest cubic metre. b) Determine the vol time of Gurjit s bin, 7, equipment on her deck. She has 1 m1 of tenth of a metre. with maximum volume, to the nearest a) Determine the dimensions of the bin plywood available. bin with a lid to hold swimming pool toy s and 7. Gurjit is building a square-based prism storage is greater than the volume of the original square-based prism. c) Calculate the volume of the prism in part b) to verify that it to the nearest tenth of a metre. maximum volume. Round We dimensions prism with the same surface area but with 1.m 1.m b) Determine the dimensions of a square-based of the square-based prism. 0.8 mu 6. a) Determine the surface area and the volume c) Calculate the volume of the box in part bj to to the nearest tenth of a centimetre. prism box with the same surface area but with maximum volume. Round the dimensions b) Determine the dimensions of a square-based the square-based prism box shown. verify that it is greater than [lie volume of the box in part a). 1cm Li-- /Rcm j6 cm rn 5. a) Determine the surface area and the volume of volume that can be made with 700 cm of cardboard. [lie dimensions of the square-based prism box with maximum. Use a table or a spreadsheet to conduct an investigation to find Connect and Apply cubic unit.. Determine the volume of each prism in question, to the nearest d) 100 m c) 750 cm b) 00 m volume for cacti surface area. Round the dimensions to tile nearest. Determine the dimensions of the square based prism with maximum

9 ass. Coiñmunicate Your Understanding - l * 508 MHR Chapter 9 crossing by a miniature Hebrides Islands of Scotland. l..abrador, and landing in the Island, Newfoundland and flight taking off from Bell to make the over 00 km unpiloted airplane took 6 h on August 5, The robotic airplane occurred The first trans-atlantic rn Did You Know? c) Describe any assumptions you have made in solving this problem. ferry tank is to be made from 8 m of aluminum, What is the can hold. Hint: 1 m = 1000 L. Connect and Apply b) Determine how many litres of liquid fertilizer this container volume. Round the dimensions to the nearest tenth of a metre. a) Determine the dimensions of the container with maximum fuel tanks are often carried in the cabin of the plane. These extra fuel of sheet metal. 7 m of metal is available.. A fertilizer company wants to make a cylindrical storage container maximum volume of fuel that it can hold, to the nearest cubic metre? Many European businesses buy aircraft manufactured in North America. To make the flight home across the Atlantic Ocean, extra tanks, called ferry tanks, must be as light as possible. A cylindrical nearest cubic unit.. Determine the volume of each cylinder in question 1. Round to the c) 15 cm d) 600 mm a) 100 cm b) 10 in nearest hundredth of a unit. volume for each surface area. Round the dimensions to the 1. Determine the dimensions of the cylinder with the maximum For help with questions I and. see the Exampie. a a p a I * a a a pal Practise at. a - a sat designed in other ways? might these glasses be given surface area. Why are designed to have the Not all drinking glasses your answer. the same surface area. Which cylinder has the greatest volume? Explain greatest volume for a Cylinder C. Cylinder B e These cylinders have Cylinder A the maximum volume of a cylinder, given its surface area. Describe a situation where it would be necessary to find

10 Cylinder dianieter. That is, h = d or h = Zr. and a height exist that produce the minimum surface area. For a cylinder with a given volume a radius The minimum surface area for a given volume Key Concepts of a cylinder occurs when its height equals its 9.6 Minimize the Surface Area of a CylinderS MHR 51 it? Round the dimensions to the nearest tenth of a centimetre. dimensions be to minimiie the amount of material used to make. A cylindrical can is to have a volume of 50 cm. What should its nearest square unit.. Determine the surface area of each cylinder in question I to the c) 5 cm d) m a) 100 cm b) 1 m of a unit. area for each volume. Round the dimensions, to the nearest tenth 1. Determine the dimensions of the cylinder with minimum surface For help with questions I to, see the Example. Practise 5, Cylinder C surface area? Explain your cylinder has tile least. the same volume. Which O These cylinders all have CylinderA the minimum surface area of a cylinder, given its volume. answer.. B 0 Describe a situation where it would be necessary to find Communicate Your Understanding will be twice that value! or r. solving the formula V = r for r, and the height surface area for a given volume can be found by The dimensions of the cylinder of minimum

11 Round the dimensions to the nearest tenth of a centimetre. a) Determine the dimensions of the can that requires the least material.. A cylindrical gas tank is designed to hold 5 L of gas. Connect and Apply 51 MHR Chapter9 F and high speed. supports three data-transfer version.0. This version rates: low speed, full speed, current specification is at standardized by the USS lmplenienters Forum, The The design of the USB is rn Did You Know? University in Sudburv. mathematically. Justify your answer amount of heat loss? to minimize the think it was designed the photo. Do you Ontario, is shown in at Laurentian 8. Chapter Problem Talia is shipping USB (universal serial bus) cables 9. A cylindrical building box or a cylinder for the cables? Justify your answer mathematically. is as cost efficient as possible. Should she use a square-based prism to a customer, She needs a container with a volume of 500 cm that designed in other ways. least amount of material. Give reasons why the cans might be 7. Many of the cans found in our homes are not designed to use the b) If aluminum costs /cm, find the cost of the aluminum amount of aluminum. Round the dimensions to the nearest tenth of a centimetre. a) Determine the dimensions of the can that requires the least to make 1 cans. 6. A cylindrical can must hold 75 ml of juice. the nearest tenth of a centimetre. container with volume 1 L that has minimum heat loss. Round to surface area must be minimized. Find the interior dimensions of the transport hot beverages. To keep heat loss to a minimum, the total 5. Wade has been asked to design an insulated cylindrical container to b) Describe any assumptions you made in solving this problem.

12 9.1 Investigate Measurement Concepts, pages a) the dimensions of various rectangles with a b) Answers will vary Example: Rectangle Width (units) (units) (units) pages 8 90 The greatest area can be achieved by using a circle. Length Perimeter Area (square Perimeter and Area Relationships of a Rectangle. units) perimeter of units 7. b) triangle: equilateral with each side 1 m, c) Yes. Difference shapes allow for different areas. area 6.5 ni hexagon: each side m, 6 area 9.6 m1 circle: radius nf 5.7 m, urea m reclangle: each side 9 m, area ill m 578 MHR Answers 6. 6 m will be the mast economical. the greatest area for the same amount of fencing.. 5 S Rectangle Width S Rectangle Width 6 (units) Length (units) Area (square Perimeter Rectangle a) Width (m) (m) 16 6 (m) 16 0 Length Perimeter Area (m) c) Rectangle (a square) with dimensions m by In 5. A rectangle with dimensions m by m encloses be stored in it. material used to construct the sited and what will b) The greater the perimeter, tile more expensive the d) Answers will vary. Example: The quality of the shed; (he smaller the perimeter, the lower the cost. units) (units) nf 1 square units using a geoboard of one, and find the necessary length. Start with a width of I unit and increase by intervals different rectangles with aa area of 1 square units. two pith be 1 unit and use an elastic band to make. a) the dimensions of various rectangles with an area b) Answers will vary. Example: Let the space between (units) (units) (units) units) Length Perimeter Area (square of 0 units. and decrease the length by the same amount to length. Then increase the width by one tnethpick perimeter of 0 units using toothpicks toothpick as the width and nine toothpicks as the construct a series of rectangles with a perimeter a) the dimensions of various rectangles with a b) Answers will vary. Example: Begin with one 10. li. Answers will van. b) 6 m by 1 at Rectangle sides 16 in; area: 56 in sides: a square with 5 6 Width (m) 5 6 (m) Length Area (m) 7 9. a) Fence Used (m) 6m flsm Gm area: 16 m 8m sides each Cm; area 6 In 8. b) extra 789 m 8 m 1m m by 1.5 rn 1. m by 1.6 m. a Answers may vary. Example: 1 m by m,. h) No. 0,5 m cannot be created using -rn barriers. 1. a)smhysm c) 1.5 m by 1.5 m d) 0.75 m by 0, a) 0.5 m by 0.5 m a) 19Gm a) extra 196 m b) 1.5 m by 1.5 m c) 0.5 m m 8 by m b) 78 m 7. sides: a square with sides: a rectangle 8 m by 16 m; c) in The greatest area, 00 m, is enclosed when the length and width are each 0 m. 56 m 8 m m b) a m by 9 m

13 - I I - Diagrams AI m by 5.9 in 1. 5mby 1Dm 15. an equilateral triangle with side length 17.cm 16. a square with side length 1.1 cm 17. Yes. A circle has greater area, 5.8 cm Each field is 1.7 m by 6.5 rn 9. MInimize the Surface Area of a Square-Based Prism, pa ges B,C,A. a) 8 cm by 8 cm by 8 cm b) 10cm by 10cm by 10cm c) 9.1 cm by 9.1 cm by 9.1 cm d) 10.6cm X 10.6cm X 10.6cm. a) 8 cm b) 600 cm c) 97 cm d) 67 cm. cube with side length 1.7cm 5. a) 15.9cm by 15.9cm by 15.9cm b) Answers will vary. 6. a) 9.09cm by 9.09cm by 9.09cm b) 95.8 cm GIn 8. a) 17.1 cm by 17.1 cm by 8.5cm b) different c) The lidless box requires less material. 9. a) cube with side length 5.8cm b) Answers will vary. Example: Cubical boxes are harder to hold and the cube would he very small. c) Answers will vary. 10. Answers will vary. 11. Try to get the square-based prism to be as close to a cube in shape as possible. The dimensions are 5 by 5 by. 1.a) / / F/ F i b) This is the closest that boxes can be stacked tn form a cube. c) Answers will vary. Example: No. A cube can be created to package 6 tissue boxes: length 1 box (1 X cm), width boxes ( x 1 cm), and height boxes ( x 8 cm) m by 1.6 in by 0. m; surface area 76. m cm cm 9. Maximize the Volume of a Square-Based Prism, pages I. B, C, A. a) 5 Un by 5 cm by 5 cm b) 0 In by 0 m by 0 m c) 11.cm by 11.cm by 11.cm d) 1.1 m by 1.1 In by 1.1 m. a) 15 cm b) 0000 In 105 cm d) 80 in. 10.8cm by 10.8cm by 10.8cm 5. a) 016 cm; 518 cm b) 18. cm by 18. cm by 18. GIn c) 618 cm a) 6.7 in; 1.15 m b) 1.1 m by 1.1 m by 1.1 m c) 1.1 m a)1.mbyl.mbyl.m Warn a) 0. cm by 0. cm by 0. cm b) 890 cm c) 7790 cm d) Answers will vary. There is no empty space in the bo.tlle DVD would fit into the cube with enough room around the edges for the shredded paper. The shredded paper is tightly packed. a) 69.cm by 69. cm by 69.cm b) will var c) Answers will vary. Assulne that Dylan cuts the wood c:arefully to not waste any pieces. 11. a) H-t b) 1000 cm c) Answers will vary. I I I J I I d) cm e) Answers will vary. Example: No loss duo to cuts. 9.5 Maximize the Volume of a Cylinder, pages a) ii = cm, r = 7.98cm b) 11 = 1.6 m, r = 0.7 m c) h = 5.16 cm, r =.58cm d) h = 6.86 mm, r = 18.mm. a) 19 cm1 c) 108 cm b) m d) 9 mm. m. a)r=.0m,h=.om b)5065l c) Answers will vary. Example: no metal will be wasted in the building process, ao metal is being overlapped 5. a) 1 cm b) 60 CDs c) Answers will vary. Example: only tile dimensions of tile CDs need to be considere; no extra space is left for the container s closing mechanism, tile plastic container has no thickness, 6. a) Answers will vary. Example: Adjust the surface area fonnula for the new cylinder, isolate the height and run a few trials using a spreadsheet to find the maximum volume, b)h 7.acm,r 7. cm 7. a) Answers will vary. b) cylinder: r = 11.&cm, volume 9018 cm; square-based prism: s 0cm, volume $000 cm Answers MHR 579

14 d) The sphere has the greatdst volume. Yes, this will e) For a given surface area: always be the case. square.based prism cm c sphere cm; cylinder c&; Rectangle Im) (m) (m) (m) h = 0.60 cm, square-based prism: s = 18.6cm Width Length Perimeter Area 1. a) 580 MHR Answers 1. s = 6.67 cm, h = 10cm 15. r. cm, h = 11.95cm 16. r = 6.91 cm, h 19.5cm 11. a) r = 7.8 cm, h = 7.8cm 1. a) Answers will vary. 1. a sphere with volume 0 18 ci& sphere 8.1 cm; The sphere has the least needed is used to enclose the required volume so a cube will have a surface area of 78 cm. A cylinder c) Answers will vary. Example: The only cardboard be harder to use, to handle, to carry, or to store. is mere cost efficient. to use cylinders with the optimum volume. They may 7. Answers will vary. Example: It is not always practical 5. r = 1. cm, Ii =.8cm 6. a)r.9 cm,h = 7.8cm 8. A cylinder will have a surface area of 9 cm, and 9. No, because the cylindrical shape is taller than its will be needed to enclose the volume. diameter. However, there is a large glass area which would encourage solar heating, thdre is no wastage. surface area. b) Answers will vary. Example: No extra material b) $. b) 576 cm b) prism 690 cm, cylinder 55.7 cm,. =. cm, = 8.8 cm F. a) r 9. cm, I, = 18.6 cm z) 05 cm. a) 6 cm b) 5 m d) 15 m b) r = 0.5 m, 1, = 1.0 m d) r 0.9 cm, h = 1.8 cm c) r =. cm, h = 6.6cm 1. a) r 5.8 cm, = 11.6cm 9.6 MInimize the Surface Area of a Cylinder, pages r 6.5 cm, h = 9.cm b) r = 0.6 m, h = 0.6 m 9. a) r = 0. m, h = 0.6 m of a square-based prism volume of a sphere> volume a cylinder > volume b) sphere: r 1.6 cm; cylinder: r = 10.0 cm, 8. a) Answers will vary. Review, pages cm by 1.1 cm by 1.1 cm 1. r = 6.18 cm, h = 1.6 cm, volume 18 c& four such pieces fit. from a 60cm by 0cm piece of cardboard because only 11. It is not possible to cut six 1.1cm by 1.1 cm pieces could save on packaging costs. than the surface area of the box. The manufacturer cylinder that contains the same volume will be less 7. a) 10cm by 10cm by 10cm gain speed. b) Answers will vary. Example: A square ice rink may b) Answers will vary. Example: The surface area of a. a) 900 m b) 1800 m S. a). m by.1 m cm by 9.6 cm by 9.6 cm 8. 1 cm m by 0.58 m by 0.58 m. 1 m by 1 m will be required. has the least perimeter. Thus, fewer edging bricks not be best as people want longer straight paths to c) m by m because for the same enclosed area, it Rectangle m) ciii) (m) (m) Length Perimeter Area H Width i r I IIiIII; b) c) 10 by 10 because it has the greatest area are integers b) 10 possible rectangles wheo the side measurements