Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 210 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 191 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 156 degrees.

The problem is based on Newton's law of cooling.

According to Newton's law

dQ / dt = k (T - T₀),

dT / dt = k' (T - T₀); dT / dt is rate of fall of temperature.

T is average temperature of hot body, T₀ is temperature of surrounding.

In the first case rate of fall of temperature = (210 - 191) / 2.5

= 7.6 degree / s

average temperature T = (210 + 191) / 2

= 200.5

Putting in the equation

7.6 = k' (200.5 - 64)

k' = 7.6 / 136.5

=.055677

In the second case : ---

In the second case, rate of fall of temperature = (191 - 156) / t

= 35 / t, t is time required.

average temperature T = (156 + 191) / 2

= 173.5

Putting in the equation

35 / t =.05567 (173.5 - 64)

t = 5.74 minute.