Set 19 - More Optimization Learning Objectives: You should be able to solve optimization problems which require using the derivative rules we've learned. As always in optimization problems, remember that finding a critical number is not the end of the problem; you still need to show that you have the absolute minimum or maximum! Reading: Boelkins 3.4 or Gottlieb 10.1 1. You have just invented a new peanut butter guacamole dip, and you stand in front of the Science Center to sell your product by the jar. Somehow, a rumor gets started (not traceable to you) that your dip cures baldness and acts as a sedative for younger siblings. Sales take off. At a price of $2.00 per jar, you can sell 500 jars a day. For every quarter that you increase the price, you sell ten fewer jars. Assuming that your fixed costs are $200 per day (for your patent lawyer) and the cost per jar is $1.00, determine the price for which you should sell your dip in order to maximize profit. 2. Frankie has decided to plant a vegetable garden in her backyard. She decides to make it a rectangle, with one side along the back of the house; the other 3 sides need fencing. The fencing costs $5 per foot, and Frankie also needs to buy top soil for the garden, which costs $2 per square foot of area. If Frankie would like the garden to have as large an area as possible and she has $200 to spend on it, what dimensions should she make the garden? Be sure to justify that your answer actually maximizes the area of the garden. 3. A dog running around on the beach spots an interesting stick in the water, 800 meters ahead and 500 meters off-shore. stick stick 500 m dog 500 m ocean beach dog 800 m x ocean beach The dog gets to the stick by first running x meters along the shore and then swimming directly toward the stick. Suppose that the dog runs at a speed of 5 meters per second and swims at a speed of 1 meter per second. What value of x minimizes the amount of time it takes the dog to reach the stick? In fact, there is evidence that dogs in this situation actually take something close to the fastest path! To read more, see the article \"Do Dogs Know Calculus?\" by Timothy J.Pennings in the May, 2003 issue of The College Mathematics Journal. You can access it online at http: // www. jstor. org. ezpprod1. hul. harvard. edu/ stable/ 3595798 Tips: 1 Finding the critical numbers will involve solving an equation with a square root. Here's a strategy for dealing with this sort of equation: to get rid of the square root, isolate it on one side of the equation, and then square both sides of the equation. (In this particular problem, you should get a nice integer answer.) The function you're optimizing will involve some squares, like (800x)2 ; the algebra will be easier if you don't expand these. 4. Mario is making a wacky picture frame. He decides that his frame will be rectangular with a diagonal of length 10 inches. The vertical sides of the frame will be metallic, while the horizontal sides will be plastic; each side needs to be at least 1 inch long. The cost of materials for the metallic sides is 4 cents per inch, while the cost for the plastic sides is 3 cents per inch. (a) Luigi suggests that Mario make the vertical sides 3 inches long. If Mario does this, how much will the frame cost? Round your answer to the nearest cent. (b) What dimensions will minimize the cost of Mario's frame? (c) How much cheaper is it for Mario to use the optimal dimensions you found in (b) than the dimensions suggested by Luigi? 5. Johanna is building a window box for her herb garden, which will be shaped as in the following picture: 2 As shown, the window box is 2 feet long, and its ends are shaped like trapezoids with bottom side 0.5 feet and angled side 1 foot: 1 0.5 What angle will maximize the volume of the box? (Hint: When solving for the critical points, it is convenient to rewrite cos2 as 1 sin2 .) 6. Preparatory Problem on xx . Note that the problem mentions that the derivative of 3x is 3x ln 3; you'll see why in class as soon as you submit this Problem Set! Let f (x) = xx , defined for x > 0. (a) Fill in the limit definition of f 0 (2): f 0 (2) = lim h0 2 . (b) Use numerical methods to approximate f 0 (2). (In (a), you expressed f 0 (2) as the limit of some expression as h 0; to approximate f 0 (2), you can plug in a value of h which is very close to 0.) (c) If we mistakenly thought f (x) was a power function like x3 , we might think that f 0 (x) = x xx1 . Refer to your answer to (b) to show that f 0 (x) 6= x xx1 . (d) If we mistakenly thought f (x) was an exponential function like 3x , we might think that f 0 (x) = xx ln(x). Refer to your answer to (b) to show that f 0 (x) 6= xx ln(x). Next time, we'll see a systematic method for differentiating functions like this. 7. (Optional extra credit, 5 points) A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 8. (Strongly recommended) Attach a study guide for this problem set! 3