IB Math SL Year 2 Name: Date: 8-3: Optimization in 2D Today s Goals: What is optimization? How do you maximize/minimize quantities using calculus?

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1 Name: Date: 8-3: Optimization in 2D Today s Goals: What is optimization? How do you maximize/minimize quantities using calculus? What is optimization? It involves finding the or value of a function subjected to a. Constraint - General Approach to solving optimization question using calculus: 1. Read the problem at least (carefully) and draw a, if appropriate 2. Identify your givens and what you are looking for 3. Assign a variable for each quantity involved in the problem. 4. Write an equation involving the quantity you are trying to make as or as as possible (Equation to be ) 5. Write an equation that involves the restrictions to the problems ( equation) 6. Use the equation to rewrite the equation to be optimized so that it is only in terms of variable 7. Find the points (derivative = ) 8. Determine the max or min value Key Note: How can we recognize that a question involves this optimization process?

2 Let s Try It! Example 1) Find the dimensions of a rectangle with a perimeter of 100 ft whose area is as large as possible. Steps 1. Identify what we will optimize: Solve 2. Identify constraint: 3. Sketch: 4. Write an equation of the quantity to optimize : 5. Write an equation for the constraint, only in terms of one variable. 6. Use constraint equation, to rewrite: 7. Find critical values. 8. Answer the question.

3 Let s Try Another 2. We need to enclose a rectangular field with a fence. We have 500 feet of fencing material and a building is on one side of the field and so won t need any fencing. Determine the dimensions of the field that will enclose the largest area. You Try! 3. You have 40 feet of fence to enclose a rectangular garden along the side of a barn. a. What are the dimensions that will maximize the area of the garden? b. Hence, what is the maximum area that you can enclose?

4 4. A rectangle has width x metres and length 30-x metres. a) What are the dimensions that will maximize the area of the rectangle? Pro Tip: Notice in this problem, there is no constraint. How can we still use steps of optimization to find the maximum? Think back to last unit. b) Find the maximum area of the rectangle. c) Find the value of the perimeter of the rectangle whose maximum area you found in part b.

5 Begin Flipped Video Lesson (prep for tomorrow) Ready? Scenario: It will take 2400 cubic inches of packing peanuts to fit in the following box. It will take 1120 square inches of wrapping paper to cover the box (Without any overlap) a) What information tells you about the volume of this box? b) What information tells you about the surface area of this box? c) What is surface area? Surface Area The surface area of a 3-Dimensional solid is... Here s an example: Find the surface area of a rectangular prism whose length is 6 meters, width is 4 meters, and height is 5.

6 *Formula Booklet One Tricky Surface Area: Cylinder

7 You Try! 1. Consider the rectangular prism to the right. a. Calculate the surface area and show all work. b. Calculate the volume. 2. Consider the cylinder to the right. a. Calculate the surface area to the nearest tenth of the square centimeter. Show all work! b. Calculate the volume to the nearest tenth of a square centimeter.