Question

Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT/dt=k(T-38) where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.

To the nearest degree, what is the temperature of the lemonade after an additional 30 minutes?

Answer 1

dT/dt=k(T-38)
ln(T-38) = kt+C
T=38 + Ce^(-kt)
Since T(0)=75, 75 = 38+C, so C=37
Since T(30)=60, 60 = 38+37e^(-30k), so k = -ln(22/37)/30 ~= -0.017329
T(60) = 38+37e^(-0.017329*60) = 51.1°F