Mathematical Modeling I Test #1

Mathematical Modeling 2001

Test #1.

1. Suppose that beginning on your birthday in the year 2000, you receive money at the rate of $10 / day, which you immediately invest into an account paying annual 8% interest starting from the day you receive it. After earning interest at rate r for t years, an initial amount P of money will grow to value P e ^ ( r t ). Note that an 8% rate is 8 per hundred, or .08.

2. Suppose you are driving at 60 mph down the highway and realize that you have to stop and take 10 seconds to secure part of your baggage in the back seat. You brake to a stop, changing velocity at a constant rate and requiring 10 seconds to reach a complete stop. You take 10 seconds to secure the load, then accelerated uniformly back to 60 mph, requiring another 20 seconds to do so. How much time did you just lose?

3. A tree has a number of bugs on it equal to the number of days until your next birthday.It is in the middle of a long line of bugless trees. Every hour there is a 20% transition of bugs from each tree to each of its neighbors.

4. Using Pascal's Triangle determine the probabilities of ending up 10, 8, 6, 4, 2 and 0 steps away from your original position on a random walk of 10 steps. Explain what the whole analysis has to do with coin flips, and what Pascal's Triangle has to do with coin flips.

5. Set up a transition matrix for a sane-demented model in which each transition sees 13% of the Sane become Demented and 19% of the Demented become Sane.

6. The plants in a certain garden are arranged in four concentric circles, i.e., circles with a common center. Plants are spaced at 1-foot intervals, and the radius of each circle is 10 feet greater than the next one inside it. The first circle has a 10-foot radius. In any given time interval, 20% of the bugs in one circle will move to the next circle out from the center, and 20% to the next circle in toward the center. Note that this refers to the total number of bugs in each circle, not on each plant. Each plant initially has 50 bugs.

The bugs on the innermost circle have no circle inside them so they don't move inward. 20% of the bugs on the outer circle get lost.

7. A container has total height 100 cm, and is in the shape of a cone. The diameter of the cone is at every point 1/10 the height of that point above the apex of the cone, which rests on a table. There is a hole .3 cm in diameter at a point 10 cm above the apex, and water exits freely from the hole. The cone is initially full to the brim. If y represents the depth of water above the hole, then the exit velocity of the water from the hole is v = `sqrt( 2 * 980 * y), where y is in cm and v is in cm / sec.

8. A random walker starts from the origin of an x-y coordinate system and takes three random steps in the x direction, then three random steps in the y direction. What distances from the original point are possible, and what is the probability of each? Note: don't forget about the Pythagorean Theorem.

9. A mass is suspended by a rubber-band system, and when it is x cm from its equilibrium position its velocity changes a rate v ' = dv / dt = -40 cm / sec * `sqrt( | x | ). The mass is initially at rest at a distance of 4 cm from equilibrium.

Repeat this process until the mass has reached its maximum displacement on the other side of the equilibrium position.

10. Use the program Kinmodel in the simulations folder of the comm folder to obtain 20 randomly chosen x kinetic energies. Find the mean and standard deviation, and determine the percents that lie within 1 standard deviation, and within 2 standard deviations, of the mean. How well do your percents agree with the prediction of a normal-curve model?

tonight:

trap graph normal curve

cellular automata

adiabatic expansion?

derivatives of polynomial and exponential functions, antiderivative, moving both ways

prepare for weekend assignment

experiment: pendulum linearity using calibrated rubber bands

do some experiments with interfaces????

10-trial coin tosses

have brains grown to point where we can make some succinct statements about distribution of mean, derivatives and antiderivatives, diff eqs and the functions that come from them, fund thm etc. and have them understood? can we go into DERIVE and do some of this stuff?

Mathematical Modeling 2001

Assignment 0702

1. The velocity of an automobile is given in meters / sec by the function v(t) = 3 t + 1, where t is clock time in seconds.

2. The velocity of an automobile is given in meters/sec by the function v(t) = .4 t^2 + 3 t + 1. Sketch a v vs. t graph and locate the t = 4, t = 7 and t = 10 points of the graph.

3. The normal curve is given by the function N(z) = 2 / sqrt(2 `pi) * e^-(z^2 / 2). For a graph of N(z) vs. z, find and plot the graph points corresponding to z = -2, -1, 0, 1, 2. Sketch the trapezoids corresponding to these points, and label heights, slopes and areas according to the conventions of the preceding problems.

 

4. There are 10 Christmas trees is a circle. Every two seconds, each tree looks at each of its nearest neighbors to see whether their lights are on or off. If exactly one of its neighbors has its lights on, it turns its light switch to the 'on' position. Otherwise it turns its light switch to the 'off' position.