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  is a classic optimization problem that uses trigonometric functions to compute an optimal volume of a trough by varying the angle of the sides. 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 question examines two optimization problems from classical Calculus that involve trigonometric functions and their understanding. In each case, a diagram is provided in the lab to help you visualize the cross-sectional area of the trough. The cross-sectional area is a trapezoid. (You may want to review the area of a trapezoid in wikipedia.) For Trapezoid A, you can readily find the value of x, then the height of the trapezoid is associated with cos(q). Use your definition of cos(q) to find height as a function of x and cos(q). One base of the trapezoid is x, while the other is x plus the other legs of the triangular regions on either side of the central rectangle. The legs of these triangular regions are associated with sin(q), so use the definition of sin(q) to find the length of these legs as a function of x and sin(q). This will give you the formula for the cross-sectional area, then the volume for this trough is simply the area of the trapezoid times the length of the metal strip. Clearly, the only variable in the area function is q, so graphing the area on the given domain is relatively simple. The optimization of this volume is a problem that you can do either on paper or by taking advantage of Maple and its ability to differentiate.

For Trapezoid B, you cannot simply divide x into three equal pieces. You must start by finding both x(q) and the height h(q) (taking advantage of the definitions of the trigonometric functions). After doing this, you use the formula for the area of a trapezoid and to find the volume of the trough. In this case, the volume of Trough B is a more complicated expression of q, so you will almost certainly want to use Maple to optimize the volume.

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.)