4. According to Newton’s law of cooling, if an object at temperature T is immersed in a medium having the constant temperature M, then the rate of change of T is proportional to the difference of temperature M − T. This gives the differential equation dT dt = k(M − T). (a) Solve the differential equation for T. (b) A thermometer reading 100◦F is placed in a medium having a constant temperature of 70◦F. After 6 min, the thermometer reads 80◦F. What is the reading after 20 min?

a) The solution of the differential equation is .

b) The reading after 20 minutes is approximately 70.770 ºF.

Explanation:

a) Newton's law of cooling is represented by the following ordinary differential equation:

(Eq. 1)

Where:

- Rate of change of temperature of the object in time, measured in Fahrenheit per minute.

- Time constant, measured in minutes.

- Temperature of the object, measured in Fahrenheit.

- Medium temperature, measured in Fahrenheit.

Now we proceed to solve the differential equation:

(Eq. 2)

Where:

-Time, measured in minutes.

- Initial temperature of the object, measured in Fahrenheit.

b) From (Eq. 2) we obtain the time constant of the cooling equation for the object: (, , , )

The cooling equation of the object is and the temperature of the object after 20 minutes is:

The reading after 20 minutes is approximately 70.770 ºF.