Solved Final Exam - Model Building in Applied Mathematics | MATH 360, Exams of Mathematics

Download Solved Final Exam - Model Building in Applied Mathematics | MATH 360 and more Exams Mathematics in PDF only on Docsity! Math 360 Final Exam with hints for solutions, May 4 1999 Spring 2001 students: We covered a slightly different set of topics this year. You may disregard all references to Euler’s method (below). Also we didn’t have much time to work with phase diagrams for systems of ODEs. 1. The state of Illinois is going to build a train line with four stations, one each in Quad-Cities, DeKalb, Chicago, and Rockford. The intended configuration is a “T” shape running west-to-east (Q − D − C) with a northward spur from DeKalb to Rockford. A parts depot is to be built somewhere on the main east-west line only but it will service all four repair stations. Except for the non-linear configuration, all the other assumptions made in our class model will be made here as well. Assume the distances from DeKalb to points Q, C, and R are, respectively, 100 miles, 60 miles, and 40 miles. Assume Q must be visited once per year, R five times, D three times, and C eight times. Develop a model and use it to decide which point on the main line gives a loca- tion for the parts depot which results in the lowest annual cost. (Your model should be sufficiently robust that it is clear how this location depends on the numbers of visits.) Let us position the parts depot a distance x from the Quad Cities, and compute the annual cost of maintaining the depot there. The cost is assumed proportional to the distance travelled, which is the sum of the products (visits to city per year)x(distance to city). If x ≤ 100, then this last is 1 · x + 3 · (100 − x) + 5 · (140 − x) + 8 · (160 − x) which simplifies to −15x+constant. Clearly this is minimized by maximizing x, i.e. taking x = 100. If instead x ≥ 100, the sum is 1 · x + 3 · (x − 100) + 5 · (40 + x − 100) + 8 · (160− x) which simplifies to +1x+constant. Clearly this is minimized by minimizing x, i.e. taking x = 100 again. In either case we see that DeKalb is the optimal location. Note the the decision depends only the number Ni of visits to the four cities; if the ‘‘8’’ is changed to a number greater than ‘‘9’’, the very same analysis shows Chicago is a better location. 2. Assume that all text books are geometrically similar, including the portion of the page filled with print. However, the size of the type is constant across all books, as is the thickness of the pages. Let H be the height of the book A be the area of the page N be the number of pages in a book W be the number of words on a page Use the proportionality symbol ∝ to answer the following: (a) How is A related to H? (b) How is N related to W? We take H as the characteristic length. Then A ∝ H2. The thick- ness of the book, another linear measurement, will also be ∝ H so if the pages have a fixed thickness, the number N of pages, found by dividing thes last two, is proportional to H too. Similarly Wp is, with a fixed font, proportional to the printed area of the page, which is in turn pro- portional to the total area of the page. Hence Wp ∝ H2. Combining these dependent variables, we see Wp ∝ N2. 3. Suppose the position x of an object moving along a straight line changes in time according to the differential equation dx dt = 1 t2 + x2 At time t = 0 the object is at position x = 1. Estimate its position at time t = 4. You can use Euler’s method: taking, say, h = 1 we are letting time t increase in steps of size ∆t = 1. In the first step x increases by about ∆x = (dx/dt)∆t = 1/(x2 + t2) = 1/(12 + 02) = 1 to x = 1 + 1 = 2. In the next step it likewise increases by about 1/(x2 + t2) = 1/(22 + 12) = 1/5 to x = 2+1/5 = 2.2. In the third step the increase is about 1/(2.22 +22) = .113 so x = 2.313 and in the last step we increase x by about 1/(2.3132 + 32) = .070 to 2.383. A larger number of smaller steps gives a value closer to 1.9125. A graphical solution is also possible and acceptable. 4. A manufacturing company has three plants X, Y , and Z, which produce x, y, and z units of a certain product, respectively. The annual profit from this production is given by P (x, y, z) = 6xyz2 − 400, 000x− 400, 000y − 400, 000z. The company must produce 1000 units annually. How should it allocate production so as to maximize profit? (Hint: It’s clear from the formula for P that in order to get any profit at all, all variables must be strictly positive!) We are to maximize R subject to the constraints that x, y, z ≥ 0 and x + y + z = 1000. This is a Lagrange Multipliers problem. The posi- tivity condition shows we are trying to maximize P on an open portion