This investigation is from Alastair McNaughton's book The New World of Calculus

Aim: To develop skills in discovery learning and problem-solving.

Instructions: You will need a large bowl of cold water, a test-tube of hot water, a thermometer and a stop-watch.

Put the test-tube of hot water in the bowl of cold water, and record the temperature each minute for about ten minutes.

Your task is to find the differential equation that best describes the cooling process. Begin by organising your data in a table like this:

t = time	s = temperature (°C)	= rate of cooling (°C/min)
0 1 2 3 ...

HINT You can find the values for  by subtracting consecutive values of s.

Now in order to form a differential equation which describes this data, we need to construct a graph of this information, showing  on the y-axis, and temperature on the x-axis.

You should be able to write down a differential equation. The next thing is to solve this differential equation. Finally once you have your solution, take one last reading of the temperature to check the accuracy of your method.

Further experiment

Repeat the same treatment you followed in the class experiment, using this data:

t = time	s = temperature (°C)	= rate of cooling (°C/min)
0 1 2 3 4 5 6 7 8	50 45 41 37 34 30 27 25 22

a. Find  by subtracting consecutive values of s.

b. Make a graph to show against s

c. Use a ruler to draw the line of best fit

d. Write down the equation of this line in the form  = ... s + ....

e. Solve this differential equation to find the general solution.

f. Substitute t = 0 and s = 50 to obtain a particular solution.

g. Hence calculate the temperature in 30 minutes.