Sixth night problems

1.  Give a good statement that provides a hint or an insight into something you think you know that some other people in the class might not know.  You may include pictures or diagrams.  Make the statement self-contained, so that anyone in the world with a sufficient mathematical background would understand what you are saying and would be able to benefit.

2.  Lower left on the grid is (0, 0).

Show the distribution of people on the lawn in the washer-flip diffusion experiment, where everyone started at the middle.

List the coordinates of the 12 resulting points.

3.  Two rocks are geometrically similar and are composed of identical materials.  One rock has a maximum diameter of 14 centimeters and a mass of 3.4 kilograms, while the other has maximum diameter 4 centimeters.  What is the mass of the second rock?

The first rock has 340 000 grains exposed on its surface.  How many grains are exposed on the surface of the other?

4.  Rabbits in a field increase every month by number

P * (300 - P),

where P is the population of the rabbits.  What are the stable, or equilibrium, values of the population?  That is, for what rabbit population(s) will the number of rabbits be unchanged?

5.  Sketch a trapezoidal approximation graph for the normal curve, where the z values are -3, -2, -1, 0, 1, 2 and the y values are respectively .004, .05, .24, .40, .24, .05, .004.

Find the area of each trapezoid.

What is the total area of all three trapezoids?

What is the area of each trapezoid, as a percent of the total area?

6.  If I take 1000 random snapshots of a pendulum, which is swinging 20 cm on either side of the middle of the meter stick, how many of the pictures do you think would be within 5 cm of the middle?  How many between 5 cm and 10 cm from the middle?  How many between 10 cm and 15 cm from the middle?  How many more that 15 cm from the middle?

Sketch a graph that shows the likelihood of being observed vs. the position of the pendulum.

7.  A spherical balloon has diameter 20 centimeters and is expanding at a rate of .1 centimeter per second.  At what rate is its volume changing with respect to clock time?  Note that the formula for the volume of a sphere is V = 4/3 pi r^3, where r is its radius.