Computer Lab Help 9

This page is designed to provide helpful information about the laboratory questions. You will find more details in the Lab Manual that accompanies this course. Begin this lab and every lab by introducing yourself to your partner. Determine the times when you can meet together during the week before the lab is due at your next Lab session. You should start this lab and each lab by typing the name of each team member and your computer number on the Lab Cover Page (or a copy of it).

The first WeBWorK problem asks questions about this help page and appropriate lecture material. This should help you work through the Lab more smoothly. This lab examines three problems in optimization, which should help you prepare for the lecture exam. In this Lab, the first problem looks at creating an open box from a sheet of paper by cutting the corners and folding the edges. You determine the optimal volume possible by making the appropriate cuts. The second problem considers creating a tent by cutting a square piece of fabric (in two ways) to create the optimal volume tent in the shape of a pyramid. The last problem examines predation of seagulls on clams and shows that the experiments do not lead to the same optimal solution seen in the lecture notes on crows predating on whelks.

Problem 1: This problem is one of the classic problems in using Calculus to find an optimum. You may want to take a sheet of paper and cut squares out of the 4 corners to give you some intuition into this problem. Recall that the volume of a rectangular box is the product of the length times the height times the width. The surface area will clearly be the area of the sheet of cardboard minus the area of the pieces cut out of the cardboard. By trying to cut out square corners from a sheet of paper, you should be able to figure the limits of the domain. (Obviously, one is x = 0 by not doing any cutting from the paper.)

Problem 2:This problem is another standard problem in using Calculus to find an optimal volume. You will first want to look up the volume of a pyramid and use the diagrams provided to develop intuition on converting the variables into ones required for computing the volume of the pyramid. You may want to print the figures in the hyperlink for the problem, cut out the appropriate shape, and fold them to observe relationships between key variables. Once you have determined the volume, V, as a function of x, then the Calculus portion of the problem is easily handled by hand or Maple.

Problem 3:This problem is very similar to the material in the lecture notes, where crows drop whelks in an optimal foraging manner. This question examines the foraging behavior of Glaucous-winged gulls eating butter clams. The first part of the problem uses Excel's Solver to fit the data for dropping clams to a smooth curve. Do not forget that when you are graphing this function and later the energy function, then you want at least 50 points on the curve with shorter spacing near any vertical asymptote. After obtaining the parameters for fitting the data, then the remainder of Part b should be relatively straightforward application of the optimization techniques in the lecture notes, i.e., use the lectures notes to guide your computations. The last part of the problem notes that other biological factors must be entering into the gull foraging behavior, so this part requires you to think more biologically for the discussion rather than precise mathematical formulae. (It would be an interesting project for someone to run experiments to determine the precise amount of kleptoparasitism amongst these gulls, and then find a mathematical formula (penalty function) for the loss of food to other gulls in the area.)