MPM 1DI Unit 8. Optimization
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1 RECALL: MPM 1DI Unit 8 Optimization Day 2: Optimization of Square Based Prisms Last class we looked at a) how to opmize (maximize) the area of a rectangle with a given perimeter & b) how to opmize (minimize) the perimeter of a rectangle given its area. Success Criteria: Aer last class, you should... be able to determine the dimensions and maximum area of a rectangle given a fixed perimeter when all four sides are enclosed. be able to determine the dimensions and maximum area of a rectangle given a fixed perimeter when only three sides are enclosed. be able to determine the dimensions and minimum perimeter of a rectangle given a fixed area.
2 Have you achieved last day's success criteria? ENTRANCE CARD: Test your understanding by answering the following quesons What is the smallest perimeter possible for a rectangle with an area of 36 m 2? 2. What is the largest area possible for a rectangle with a perimeter of 28 m? 3. Brandon wants to create an enclosure for his puppy in his backyard. He wants to use the side of the house as one side of the enclosure, so he only needs to fence 3 sides of the enclosure. He has 12 m of fencing. What are the dimensions and the area of the rectangle with maximum area?
3 TODAY: Success Criteria: By the end of today's lesson can you... determine the minimum surface area of a square based prism given a fixed volume. determine the maximum volume of a square based prism given a fixed surface area.
4 Investigation A : How can you compare the surface areas of square based prisms with the same volume? Let's invesgate Use 16 interlocking cubes to build as many different square based prisms as possible with a volume of 16 cubic units. 2 Calculate the surface area of each prism and record your results in the table. Length Width Height Volume Surface Area What are the dimensions of the square based prism that has the minimum, or opmal, surface area? 4. Describe the shape of this prism compared to the other prisms.
5 5. Predict the dimensions of the square based prism with minimum surface area if you use: a) 27 cubes b) 64 cubes. c) 125 cubes. REFLECT: Summarize your findings. a) Do any relaonships exist between the length, width, and height of a square based prism with minimum surface area for a given volume? b) What is the ideal shape for minimizing the surface area of a squarebased prism when given a fixed volume? c) How can you predict the dimensions of a square based prism with minimum surface area if you know the volume?
6 EX. 1. Cardboard Box Dimensions a) The Pop a Lot popcorn company ships kernels of popcorn to movie theatres in large cardboard boxes with a volume of cm 3. Determine the dimensions of the square based prism box, to the nearest tenth of a cenmetre, that will require the least amount of cardboard. b) Find the amount of cardboard required to make this box, to the nearest tenth of a square metre. Describe any assumpons you have made.
7 Investigation B : How can you compare the volumes of square based prisms with the same surface area? Let's invesgate Each of the square based prisms below has a surface area of 24 cm 2. Calculate the area of the base and the volume of each prism. Record your data in the table. Prism 1: Prism 2: Prism 3: 2 cm 3 cm 3 cm 0.5 cm 5.5 cm 2cm 2cm 1cm 1cm What are the dimensions of the square based prism that has the maximum, or opmal, volume? 3. Describe the shape of this prism compared to the other prisms.
8 4. Predict the dimensions of the square based prism with maximum volume if the surface area is 54 cm 2. REFLECT: Summarize your findings. a) Do any relaonships exist between the length, width, and height of a square based prism with maximum volume for a given surface area? b) What is the ideal shape for maximizing the volume of a square based prism when given a fixed surface area? c) How can you predict the dimensions of a square based prism with maximum volume if you know the surface area?
9 EX. 2. Maximize the Volume of a Square Based Prism a) Determine the dimensions of the square based prism with maximum volume that can be formed using 5400 cm 2 of cardboard. b) What is the volume of the prism?
10 Practice: p. 495 #2, 3, 5a, 7 & p. 501 #2, 3, 6, 7
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