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Teachers' Lounge Discussion: New examples of growth in calculus

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From: Michael Sakowski 
To: Teacher2Teacher Public Discussion
Date: 2007082708:29:24
Subject: Calculus Project - Newton's Law of Cooling

Here's a nice hands-on project for your students that I wrote up.


COFFEE TIME IS CALCULUS TIME
	
Introduction:

In this project you will compare how well your favorite coffee cup
keeps a hot cup of coffee hot.  You will solve a differential equation
for the temperature in the cup and use this to predict how long it
will take for the coffee to get cold in the cup. 

According to newtons law of cooling and heating, the rate of cooling
of an object is directly proportional to the difference of the
temperature of the object and medium surrounding the object.  This is
given by the equation

dy/dt = k(y - M)  

where dy/dt = rate of change of temperature of the substance, 
M = the temperature of the surrounding medium, 
y = the temperature of the substance at any given time t, and
k = a constant that must be solved for.

Procedure
1.Record the room temperature.  This is the temperature M from the
above equation.

2.Find a coffee cup that you wish to test.   Fill the cup with hot tap
water (or coffee).

3.Place the thermometer into the cup.  After a minute or two, record
the temperature in the cup and record the time.  Use a stopwatch if
you have one.  This is the initial temperature yo.  (Time is t = 0)  

4.Just before 6 minutes, stir the cup with the thermometer.  Then
measure the temperature of the cup at exactly 6 minutes.

5.Just before 20 minutes, stir the cup with the thermometer.  Then
measure the temperature of the cup at exactly 20 minutes.

ROOM TEMPERATURE = _______

TIME	        TEMPERATURE
INITIAL TIME     ______
 (t = 0)	

6 MINUTES	 ______

20 MINUTES	 ______


WRITE-UP QUESTIONS ARE ON THE FOLLOWING PAGE
Write-up

1.Solve the separable equation dy/dt = k(y - M) and obtain a general
solution.  

2.Use the data at time t=0 and t=6 to find the particular solution for
this experiment.(Solve for k and C.  M = room temp) 

3.Compute the temperatures at time t=20 by using your particular
solution that you just found.  

4.Compute the percent difference between the measured temperature and
computed temperature at t=20 minutes.  

	This is (measured - computed)/measured X 100%

5.Explain why the measured temperature was higher than the computed
temperature.  

6.Using the particular solution,  predict the time it will take for
the temperature in the cup to be 1 degree higher than the room
temperature.

7.Discuss ways in which your coffee cup could be made more efficient
in holding in heat.

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