3. Suppose the position x of an object moving along a straight line changes in time according to the differential equation.

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1 Math 360 Final Exam with hints for solutions, May 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 location 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 N i 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 A N W be the height of the book be the area of the page be the number of pages in a book be the number of words on a page

2 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 H 2. The thickness 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 W p is, with a fixed font, proportional to the printed area of the page, which is in turn proportional to the total area of the page. Hence W p H 2. Combining these dependent variables, we see W p N Suppose the position x of an object moving along a straight line changes in time according to the differential equation dx dt = 1 t 2 + x 2 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/(x 2 + t 2 ) = 1/( ) = 1 to x = = 2. In the next step it likewise increases by about 1/(x 2 + t 2 ) = 1/( ) = 1/5 to x = 2+1/5 = 2.2. In the third step the increase is about 1/( ) =.113 so x = and in the last step we increase x by about 1/( ) =.070 to A larger number of smaller steps gives a value closer to 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) = 6xyz 2 400, 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 = This is a Lagrange Multipliers problem. The positivity condition shows we are trying to maximize P on an open portion

3 of the plane x + y + z = So we compute the three partial derivatives and eliminate the Lagrange Multiplier λ. We get the equations dp/dx = dp/dy = dp/dz, i.e. 6z 2 (x y) = 6yz(z 2x) = 0 as well as x + y + z = Since z 0 and y 0 we must have y = x and z = 2x, so x + y + z = 1000 means 4x = 1000, x = 250, y = 250, z = 500. But it s much easier to note that if z = 1000 x y then we are trying to maximize P = 6xy(1000 x y) 2 400, 000, 000 on the open region x > 0, y > 0, x + y < 1000 of the x y plane. Taking partials with respect to x and y and discarding the factor 1000 x y in each case, we are left with 3x + y = 1000 and x + 3y = 1000, whose solution is x = y = 250; then solve for z. 5. Suppose you are doing a dimensional analysis problem involving the variables z, y, x, w, and v. You have discovered that { yw v, yz x } is a fundamental system of dimensionless products. In addition, you have some insight about the model which assures you that z should always be proportional to v, irrespective of any other changes in the other variables. Combine these two observations to say as much as you can about how z is related to w. By Buckingham s theorem, the only possible relations are of the form F (yw/v, yz/x) = 0. Solving this equation for the second variable will give an equation of the form yz/x = f(yw/v), where f is an unknown function of one variable. We may write this as z = (x/y)f(yw/v). Now, if you also know that z is proportional to v, then this right hand side must be of the form X v for some X which involves the other variables but not v. It follows that f must have the form f(u) = K/u for some constant K. Thus we conclude z = (x/y) K/(yw/v) = K xv/(y 2 w). In particular, we conclude z is inversely proportional to w. 6. It is found that a certain heavy ball, falling through water, has a velocity v which changes in time according to the equation dv dt = 4 3v in an appropriate set of units. The ball starts from rest. Compute the ball s velocity as a function of time. You should be able to show that as time increases, the ball does not move faster and faster but rather approaches a limiting velocity v. Compute v.

4 Of course this is a separable differential equation. We solve dv/(4 3v) = dt to get ln( 4 3v )/( 3) = t + C, which we solve to get v = (4/3) Ae 3t for some A. We match the initial condition v(0) = 0 to get v = (4/3)(1 e 3t ). In particular as t we see v approaches 4/3. 7. Two populations evolve according to the autonomous system dx dt dy dt = 0.4x.002xy = 0.5y +.001xy Sketch all the orbits in the first quadrant (x 0, y 0), showing the equilibria (fixed points). Do this carefully and completely, showing the direction of motion along the orbits. Explain (briefly) why the orbits have the shape you indicate. There are equilibria at the origin and at (500, 200). Divide the quadrant into the appropriate four regions and see the orbits must spiral around the nonzero equilibrium point. We have discussed in class that the orbits do in fact close the loop rather than spiral in or out. Special orbits include the two axes, the vertical axis traversed downward, the horizontal traversed left to right. There is nothing to this problem except re-creating what we did several times in class. 8. A recent sample of pizzas prepared by Pizza Hut yielded the following data on diameter (D) and weight (W ) of pizzas: D (inches) W (ounces) Now, one could argue that we should expect W D 2 (if all pizzas are geometrically similar but of a fixed thickness) or W D 3 (if all pizzas are geometrically similar, including having a thickness proportional to their diameter). Which model is better supported by the data? If you could select any model of the form W D r, what value of r would you suggest? Use your best assessment of the nature of pizzas to predict, from these data, the weight of a 24-inch-diameter MegaPizza(tm). Several answers are possible. We can look for the best possible fit to the models W = AD r both for r = 2 and r = 3, using several different notions of best. For example, the best fit in the least-squares sense can be computed as W = D 2 and W = D 3 respectively. But the former model has individual residuals as large as 7.4, whereas in the other case the sum of the squares of all eight residuals only totals 8.92.

5 We can also transform the data, plotting y = ln(w ) against x = ln(d) and fitting a straight line to these data. The best fit in the leastsquares sense has r = (and b = 4.791) so clearly the optimal r is much closer to 3 than to 2. Based on these statistical tests, and consistent with the model we can develop based on geometric similarity, we choose to accept W = AD 3 for our model, and use A = This predicts the two-footer will weigh a whopping ounces (over 7 pounds).