Mathematical Modeling 2001

1.  An inverted cone has a hole 40 cm below its apex, just below which the cone is sealed. The diameter of the cone at a point is 1/4 the vertical distance of the point below the apex. The water level is initially 25 cm above the hole. The velocity of the escaping water is `sqrt(1960 y), where y is the depth of the water above the hole.

• Approximately how long will it take the water to reach the 20 cm level, provided the diameter of the hole is 1 cm?
• Approximately how long will it take the water to reach the 15 cm level?
• Approximately how long will it take the water to reach the 10 cm level?
• Is the water level decreasing at a constant, an increasing or a decreasing rate?

2. The Normal curve is defined by 1 / `sqrt(2 `pi) e^(-z^2/2). Using a trapezoidal approxmiation graph determine the areas between z = 0, z - .2, z = .4, z = .6, z = .8 and z = 1. Summarize your work with a completely labeled trapezoidal approximation graph. Then use your results to determine the number of students from among a group of 1200 students in which the average IQ is 120 and for which the standard deviation is 15, who will have IQ's between 129 and 135.

3. Sketch and completely label a Trapezoidal Approximation Graph for the function y = .1 t^2 - 3 t + 70 for t = 0, 5, 10, 15, 20, 25. Sketch a graph of slope vs. midpoint time and also make a table, then sketch a graph of accumulated area vs. midpoint time and again make a table. Use Excel to determine the minimum degree of the polynomial that will fit the points for each table, and give the functions you obtain.

What would be the slope graph of the area vs. midpoint time graph? What would be the area graph obtained from the slope graph?

Interpret each graph provided the original function is taken to represent velocity vs. clock time.

4. Concentric spherical shells are planted with bug-producing trees, with a plant to every square meter. The diameters of the spheres are 2 meters, 4 meter, 6 meters and 8 meters. During every transition the population of bugs increases by 5%, and 10% of the bugs on each shell travel to each neighboring shell. 20% of the bugs on the outermost shell go away and get lost. There are originally 10 bugs on each tree.

• What will be the populations of the shells after 1, 2 and 3 transitions?
• What will be the populations after 100 transitions?

5. Investigate the iterates of the map m x (1-x) for various values of m, and as nearly as possible determine the value of m for which

• the graph first bifurcates,
• the graph undergoes its second bifurcation
• the graph first enters chaos

6. If 1000 individuals are concentrated in a small area, and if they begin to random-walk in the x and the y direction, each individual alternating a step to the right or lef in the x direction with a step up or down in the y direction, then after four steps what will be the distribution of distances from the original point?

7. What is the future value of a constant income stream of $100 per month at a date 10 years in the future, assuming 8% annual interest compounded continuously?

8. If stocks are up on one day, there is a 50% chance they will be up the next day, a 30% chance they will be unchanged the next day and a 20% chance they will be down the next day. If stocks are unchanged one day there is a 30% chance they will be down the next, a 30% chance they will be unchanged the next, and a 40% chance they will be up the next day. If stocks are down one day there is a 50% chance they will be down the next, a 20% chance they will be unchanged and a 30% chance they will be up the next day. Set up a transition matrix to show what happens over the next 5 days, if today there is a 50% chance that stocks are up, a 30% chance that they are down, and a 20% chance that they are unchanged. Treat this like a sane/borderline/demented situation.

9. If the rate-of-depth-change function is y = -.2 t^2 + 3 t – 15, then approximately how much depth change would be expected between t = 2 and t = 5?

10. A cellular automaton with 9 cells has the cell at the far left always on. Other than that, a cell will be on if exactly one of its neighbors is on. If the initial state has the three leftmost cells all on, then what orbit will the system fall into? Can the system achieve any other orbit?

11. The fish population in a pond changes at the rate dP/dt = .3 P ( 100 – P) / 100. There are initially 10 fish in the pond. Use this relationship to estimate as accurately as possible the number of fish in the pond at t = 2, 4, 6, 8 and 10.

12. What would be the average distance from the starting point on a 4-step random walk along a straight line? What would be the result for a 2-step random walk along the x axis followed by a 2-step random walk in the y direction?

13. What would the transition matrix look like for a row of trees on which the bugs from each tree did a 2-step random walk at each transition?

14. What is the rate-of-depth change function for depth function y = .01 t^2 – 30 t + 100?