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.

• How much would the money you receive today be worth on your birthday in the year 2010? ** for Oct 14 birthday I have 9.226 years left so today’s money will grow to $10 * e^(.08 * 9.22) = $20.92. **
• How much would the money received a year from today be worth on your birthday in the year 2010? ** will have 8.22 years to go so will grow to $10 & e^(.08*8.22) = … **
• How much would the money received 5 years from today be worth on your birthday in the year 2010? ** will have 4.22 years to go so will grow to $10 & e^(.08*4.22) = … **
• How much would the money received 9 years from today be worth on your birthday in the year 2010? ** will have 0.22 years to go so will grow to $10 & e^(.08*0.22) = … **
• Does it look like the values you are calculating are changing linearly with the number of years from today?

• How much do you think the money received in the first year since your birthday will be worth on your birthday in 2010? ** On the average the money has about 9.5 years to grow; the function isn’t linear so this is just an approximation. So the $3652.50 I got this year grows to P e^(rt) = $3652.50 e^(.08 * 9.5), approximately. **

• How much do you think the money received between your birthdays in 2005-2006 will be worth on your birthday in 2010? ** On the average the money has about 4.5 years to grow; the function isn’t linear so this is just an approximation. So the $3652.50 I got this year grows to P e^(rt) = $3652.50 e^(.08 * 4.5), approximately. **

• How much do you think all the money received in the ten years since your 2000 birthday will be worth on your birthday in 2010? ** estimate by adding up contributions from years 1-10; or note that the rate at which money accumulates is y ‘ = $3652.50 e^(.08 * (10-t)) and find the y function using combination of magic, DERVIVE, calculus. **

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? ** For the slowdown and the speedup velocity changes at a uniform rate so your average velocity is half the 60 mph. At half the velocity you only go half as far, so half the time required to slow down then to speed up is wasted. You lose half of the 30 seconds spent slowing down and speeding up, plus the 10 seconds you were stopped. **

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.

• Graph the distribution of bugs for the first 5 transitions.
• How many bugs are on the original tree at the end of each of the first five hours?

• Write these numbers as a sequence S and determine if the ratios of the sequence S ' are constant or nearly constant. If they are not constant, do they change in any predicatable way?
• Graph the number of bugs on the original tree vs. the number of hours and see how well an exponential function fits the graph, and also how well a quadratic function fits your points.

• Starting with the numbers after the first hour, write the total number of bugs on the original tree and its two immediate neighbors as a sequence S and repeat the preceding series of tasks for this sequence.

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. ** n=10 row is 1 10 45 120 210 252 210 120 45 10 1. This means that there is 1 way to take 0 ‘rights’ resulting in distance 10, 10 ways to take 1 ‘right’ resulting in distance 8, 45 ways to take 2 ‘rights’ resulting in distance 6, etc.. Following this reasoning there are 2 ways to end up 10 units away, 20 ways to end up 8 units away, 90 ways to end up 6 units away, 240 ways to end up 4 units away, 420 ways to end up 2 units away and 252 ways to end up where you started. The probabilities are therefore 2/1024, 20/1024, 90/1024, etc..

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.

• Use the matrix to determine the numbers of sand and demented after each of the first three transitions, starting with a number of sane equal to the number of days since your last birthday, and a number of demented equal to the number of days until your next birthday.
• Use algebra to determine the sane and demented populations that would result in no change to either population. ** populations become unchanging when number of sane becoming demented is equal to the number of demented becoming sane. This happens when .13 S = .19 D. So S = 19/13 D = 1.47 D, approx.. S + D = 365. Substituting 1.47 D for S, we have 1.47 D + D = 365 so 2.47 D = 365 and D = 365 / 2.47, etc., etc. **
• Write the numbers of sane as a sequence S, and show that the difference sequence S ' can be modeled by an exponential function. ** the ratios of the S’ numbers will all be the same, which implies an exponential model. If you used Excel the exponential model would have worked just fine. However, if you tried to fit S with an exponential model it wouldn’t have worked. **

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.

• After the first transition, how many bugs will there be on each of the first four circles, and how many bugs on each plant in each of those circles?

• Write the total number of bugs on all the trees after each transition as a sequence and analyze as best you can the behavior of the sequence.

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.

• At a depth of 90 cm above the hole, the cone is full. How rapidly will water be descending in the cone when it is at this depth? ** water flows out with velocity `sqrt( 2 * 980 * 90) = 420, representing 420 cm/s.

• How rapidly will water be descending when the depth is 60 cm?
• How rapidly will water be descending when the depth is 30 cm?
• Set up a series of Excel calculations to approximate depth vs. clock time using 3-second time increments. Obtain a graph of depth vs. clock time and see what sort of curve fits your data.

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^2 * `sqrt( | x | ).

** note shoulda been –40 cm/sec^2 * `sqrt |x| ) * ( x / | x| ). **

The mass is initially at rest at a distance of 4 cm from equilibrium.

• Determine its rate of change of velocity, and its velocity .1 second after being released.

• Determine as best you can the approximate change in its position during this .1 second.
• Determine your best approximation to the new position after this initial .1 second.

• Determine the rate of change of its velocity at this new position, and its velocity .1 second later.

• Determine as best you can the approximate change in its position during this .1 second.

• Determine your best approximation to the new position after this second .1-second interval.

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?