Canalysis Name: You work on the research and design team for the National Beverage Corporation (distributor of La Croix'n Sparkling Water}. You are given the task of designing a new can ofthe naturally essenced sparkling water that minimizes production COStS. 1. The current can is 12.1 centimeters tall and has a radius of 3.1 cm. Come up with the dimensions of two new cans that hold the same volumeone with a bigger radius and one with a smaller radius. (V = rzh) Original Can New Can #1 New Can #2 Radius {cm} 3.1 Height (cm) 12.1 Volume (cm3) 2. The aluminum for the side of the can costs 50.00016 per square centimeter but the thicker sheets needed for the top and bottom cost 50.0005 per square centimeter. Calculate the surface area of the three cans and then figure out the cost of producing each can. {SA = 21:?'2 + 2mh) Original Can New Can #1 New Can #2 Surface Area of Bases Surface Area of Side Total Cost 3. Maybe your classmates designed a less expensive can than you! Use a dot sticker to plot your can's radius and production cost on the class graph. Then sketch the graph below. 1 0.9 Total Cost 0.6 0.? 0.8 0.5 0.4 0.3 0.2 0.1 -- CALC MEDIC 4. Maybe we can find an equation for this curve. a. Given the radius and height of a potential design, write an equation for its production cost. b. Use the volume of the original can to rewrite the height in terms of the radius. c. Write an equation for C0\") that gives the cost of producing a can only in terms of the radius. 5. How can we use this equation to nd the least expensive can? a. For which radius is the cost of producing a can at a minimum? How do you know? b. What is the height of this least expensive can? c. What is the absolute lowest price for which we can produce a can of La Croix? -- CALC MEDIC Topics 5.10-5.11-Optimization Important Ideas: Check Your Understanding! 1. You have 200 feet of fencing to enclose a rectangular field by your house. One side of the field borders a pond and does not require fencing. What is the area of the largest field you can enclose? 2. A rectangle has two vertices on the x-axis and two vertices on the curve of y = 9 - x2. What is the maximum area of this rectangle? (Hint: Sketch a picture) 3. A sheet of cardboard with side length 60 inches is folded into a rectangular box with a lid. Four squares with side length x and two rectangular regions are discarded in order to fold the box, as shown. For what value of x is the volume of the box a maximum? (V = lwh) CALC MEDIC