Montana State University M161: Survey of Calculus 78 Section 4.5 -­‐ Optimization Part 2 Optimization Problems Maximizing Grades while Minimizing Stress 1) RECTANGLE – Largest Area?: Find the dimensions of a rectangle with perimeter 1000 ft and whose area is as large as possible. DRAW a Picture & Label Variables IDENTIFY what is to be optimized (minimized or maximized) FUNCTIONS relating info – usually a geometry formula SUB one Function into the other You need to end up with a function that has ONLY 1 Variable Compute the first derivative and find critical numbers PROVE IT! Use 1st or 2nd Derivative Test to prove Max/Min Remember Units! S. Schaefer Montana State University M161: Survey of Calculus 79 Examples 2) CANS -­‐ Betty Moore Company requires that its corned beef hash containers have a capacity of 54 cm3, have the shape of right circular cylinders, and be made of aluminum. Determine the radius and height of the container that requires the least amount of material. 3) FENCING a PIG PEN -­‐ A farmer with 500 feet of fencing wants to build a rectangular pen consisting of 4 parallel pens (as shown on the right). What dimensions will maximize the total area of the pig pen? S. Schaefer Montana State University M161: Survey of Calculus 80 4) CARDBOARD BOXES -­‐ By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, the cardboard may be turned into an open box. If the cardboard is 16 inches long and 10 inches wide, find the dimensions of the box that will yield the maximum volume. 5) PAGE MARGINS -­‐ A rectangular page is to contain 24 in2 of print. Top/Bottom margins: 1.5 inches Left/Right margins: 1 inch What should the dimensions of the page be so that the least amount of paper is used? S. Schaefer Montana State University M161: Survey of Calculus 81 6) BUS PASS -­‐ A city’s Metro Transit Authority operates a subway line for commuters from a certain suburb to the downtown metro area. Currently, an average of 6000 passengers a day take the trains, paying a fare of $3/ride. The MTA board is contemplating increasing the fare to $3.50/ride to generate more revenue. They hire a consulting firm, and the firm’s study reveals that for each $0.50 increase in fare, the ridership will be reduced by an average of 1000 passengers a day. THUS, the consulting firm recommends that MTA stick to the current fare of $3/ride, which already yields a maximum revenue. Show that the consultants are correct. (!"#"$%" = #!"#$!% ∙ !"#$$) 7) APPLE TREES – There are 50 apple trees in an orchard. Each tree produces 800 apples. For each extra tree planted in the orchard, the amount of apples produced per tree drops by 10 apples. How many trees should be added to the existing orchard (of 50 trees) in order to maximize the total number of apples produced? (!"#$% # !""#$% = #!"##$ ∙ #!""#$%/!"##) S. Schaefer