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Optimizing Area and Perimeter of shapes

Optimizing Area and Perimeter of shapes. What is Optimization. Optimization: The process of finding values that make a given quantity the greatest (or least) possible given certain conditions. For example you can create a shape with greatest area but least perimeter. Example:.

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Optimizing Area and Perimeter of shapes

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  1. Optimizing Area and Perimeterof shapes

  2. What is Optimization • Optimization: The process of finding values that make a given quantity the greatest (or least) possible given certain conditions. • For example you can create a shape with greatest area but least perimeter.

  3. Example: • Suppose a friend asks you to help build a deck for their parents' cottage. • How can you determine the most suitable size and shape for the deck? • Oft en, there are constraints that must be considered, such as: • • the budget • • fixed perimeter • • fixed area • • natural or artificial boundaries • An expert in construction can apply measurement and geometry concepts to optimize the design of a deck or other structures.

  4. Areas of Rectangles With Fixed Perimeter • If you were given 12m of fencing and asked to fence off the largest rectangular area possible for a dog run, how would you go about it? What different rectangles could you use?

  5. Let's look at various possibilities and put the results in a table.

  6. Which rectangle has the largest area? • Which rectangle has the smallest area? • Which rectangle would you use if you were roping off a dog run?

  7. Fixed Area Rectangles In the last question, we were given a certain amount of fencing and asked to find different areas that could be fenced off. What if it was done the other way, that is, we are given a certain area to fence off, but we want to use the least amount of fence?

  8. Let's say we want to make a pet exercise area with the least amountof rope. We want the area to be 36m2. What are the possibledimensions? Let’s make another table for the length and width of a rectangle with the constraints

  9. 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 12m of fencing. a) Do you think he would be able to enclose more, less or the same amount of area as when he did not have the house as one side? b) What shape do you think will have the maximum area? c) Make a hypothesis about what dimensions will have the maximum area.

  10. e) What are the dimensions of the rectangle with the maximum area? f) Compare the results with your hypothesis. g) Compare this result with the result in Example 1 (which had the same amount of fence).

  11. Key ideas for Rectangles 1. For a fixed perimeter, the way to maximize the area of a rectangle is to create a square. 2. For a fixed area, the way to minimize the perimeter of a rectangle is to make a square. 3. Optimizing is figuring out how to get the most (maximize) good stuff like money and space and how to get the least (minimize) bad stuff like costs and effort. 4. As rectangle optimizing problems get more complicated, you need to try it out to see what arrangement is best.

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