FINAL EXAM, MATH 382

2:30 PM, WEDNESDAY, MARCH 15, 1995

Name:

One notebook sized page of notes allowed (you may write on both sides). Calculators are recommended. Show all of your work.

1. Consider the following system of equations for two populations x and y.

x′ = 2x − xy, y′ = −y + 3xy

(a) Give an interpretation in words of this system.

(b) Find the equilibrium points.

(c) Find the x − y equation of the orbits and describe them. (Are they closed curves? Do they approach certain of the equilibrium points? Does one of the populations go to 0?, etc.)

2. Consider the following system of equations for two populations x and y.

x′ = 6x − 2xy − 2x^{2} y′ = 6y − 3xy − 3y^{2}

(a) Give an interpretation in words of this system.

(b) Find the equilibrium points.

(c) Describe the orbits (don’t try to find the equations of the orbits). (Are they closed curves? Do they approach certain of the equilibrium points? Does one of the populations go to 0?, etc.)

3. A bacterial population grows according to the logistic equation, with an equilibrium density of 3 × 10^{8} cells per ml. When the population density is low, it doubles every 30 minutes. What will the population density be after 4 hours if it is initially 10^{8} cells per ml.

4. The system of equations

x′ = −ay, y′ = −bx − cxy

is a Lanchestrian model for conventional-guerilla combat in which the operational loss rate of the guerilla force is proportional to the strength of the conventional force.

a) Which variable represents the guerilla force?

b) Find the orbits of this system of equations.

5. During a steady snowfall, a man starts clearing a sidewalk at noon, shoveling at a constant rate and at a constant width. He shovels two blocks by 2 p.m. and one block more by 5 p.m. When did the snow begin to fall?

6. A baseball was thrown off of the top of the Washington Monument (555’ tall) and Michey Cochran (a Hall of Fame catcher) tried to catch it. He failed because he could never position himself under it. Suppose the ball weighs 5 oz and the air resistance is proportional to the square of the velocity of the falling ball. Suppose this proportionality constant is .00001 slug/ft. Find the limiting velocity (in miles per hour) of the baseball. You may suppose that the baseball is dropped from a great height and reaches the limiting velocity before it hits the ground. You may use g = 32 \frac{ft}{sec^{2}} and one mile = 5280 ft. (To find the limiting velocity, you don’t need to completely solve the equation.)