Calculus 2015 KetchumOptimization
Optimus Prime (get it?)
Problem Solving Strategy For Applied Maximum And Minimum Problems:
- Assign symbols to all given quantities and quantities to be determined. When feasible, draw a sketch.
- Write a Primary Equation for the quantity that is to be maximized or minimized. Check the front cover of the book or the posters on that walls for basic geometric equations and models.
- Reduce the primary equation to one having a single independent variable. This may involve the use of Secondary Equations that relate the independent variables of the primary equation. (We cannot differentiate the primary equation if there is more than one variable present)
- Think of the domain of the primary equation. Estimate the final values that will make sense.
- Optimize. Find the wanted maximum or minimum values with the use of the calculus techniques that we have learned in Chapters 2 and 3.
Example 1: Find the maximum volume
Gopro wants to design an open box that has a square base and a surface area of 108 inches for a product advertisement. This is too large for standard Gopro packaging, but it will look better on TV this way. What dimensions will produce a box with maximum volume?
Example 2: Finding minimum area
The IKEA company needs to design a new catalog. IKEA has realized that a calculus student named Jack can optimize the size of their catalog to save precious trees. The cover of the catalog will be printed on heavy paper and will contain a large picture of Jack that is 24 square inches in area. To correctly manage the FengShui of the appearance of the catalog, IKEA demands there be a margin of 1.5 inches on the top and bottom of the cover and a margin of 1 inch on each side of the cover. What dimensions of the heavy paper, and therefore catalog, will allow IKEA to use the least amount of heavy paper possible?
Is IKEA really doing this to save trees or just to save money? Was Jack used as a pawn in IKEA’s corporate structure?
Example 3: Finding minimum length, two parts
Two circus tent posts, one 12 feet high and the other 28 feet high, are 30 feet apart. A circus worker needs to use one long piece of rope to support each of the posts. He will attach each end of the rope to the top of each post and then drive a stake into the ground to hold the rope to the ground between the posts. His circus boss gave him a spool of rope with 48 feet of rope on it.
a. Does the circus worker have enough rope to support both tent posts?
b. Whether he does or does not have enough rope… where should he stake down the rope between the poles?
c. If he does not have enough rope (hint hint), what is the minimum amount of extra rope he needs?
Circus Part Deux: Revenge of the Circus Rope
Two posts are both 12 feet high. The circus worker does not have any form of distance measuring device on him. How can he find where to stake the rope to the ground between the two posts to use the least amount of rope? Use logic and prior knowledge of distance to figure this out. (Remember MacGyver?)
Example 4: I won’t give this one away so easily
Logic.Experience.Knowledge.Science. USE THEM!
Southern California Edison has four miles of copper wire to surround two new housing developments in the middle of nowhere. Desert Meadows is in the shape of a square and Sunny Plains is in the shape of a circle. Although SCE has heaping stacks of money, they need to make the best use of their expensive copper materials so there fat stacks can grow evenfatter. How much of the copper wire should be used to construct the square and how much of the copper wire should be used to construct the circle to enclose the maximum area of each housing development?
If you know the answer RIGHT NOW, write it down here:
(This is one of my favorite questions that we have done so far)