2007-09-20

Today's nonsense
- A kid going through Dakota's phone, changing the wallpaper, backgrounds, tones, calling people, etc.
- Administrators calling a "lock down" for paperclips or staples on the floor
- Xeroxing some slope fields
- PhysiCal

Homework due tomorrow. There are some slope fields to graph on #15 and #16.

Newton's Law of Cooling

So this is going to be about rates. The rate of change in the temperature of an object is proportional to the difference between the object's temperature and the temperature of the surrounding medium.
dT/dt = k(T-T_s) ===> ln|T-T_{s}| = C_{1}e^{kt}

If the temperature of an object is 100 degrees and is placed in a room held at 60 degrees (all Fahrenheit), how long will it take for the object to cool to 80 degrees Fahrenheit if it is 90 degrees after 10 minutes. How do you find the value of C_{1} in this case? You solve at t = 0, where you know that T = 100, and so you can find that the value of C_{1} is 40. Now you can use the other ordered pair, 90 degrees for when t = 10, so T=90 and now solve for k. And then the question wants you to solve for 't' when T=80, and so then you just need to solve normally.
C = initial difference between object and the surroundings

Let's try another one, an odd one from the same pages as on the homework page.

ex) The rate of change of y is proportional to y. When x = 0, y=4, and when x = 3, y= 10. Find y when x = 6.
dy/dx = ky
k = 0.3054

Fibanocci sequence- 1, 1, 2, 3, 5, 8, 13, 21

The rate of change of P varies jointly as J and P-3.
dP/dJ = kJ(P-3)