A mug of coffee is removed from a hot plate and placed in a room whose air…
PHYSICS · Multi-part · CBSE · Physics · Thermal Properties of Matter · Thermal Properties of Matter
Problem
A mug of coffee is removed from a hot plate and placed in a room whose air temperature is a constant 20∘C. At the moment the coffee is placed on the table (t=0 minutes), its temperature is 90∘C. After 5 minutes, the coffee has cooled to 70∘C. Let T(t) be the temperature (in ∘C) of the coffee t minutes after it is placed in the room.
(1) Write a differential equation that models the rate of change of T(t) in terms of the difference between the coffee temperature and the room temperature.
(2) Solve the differential equation to obtain an explicit expression for T(t), including a constant of proportionality k.
(3) Using the information that T(5)=70, determine the exact value of the constant k (include units).
(4) Determine the time, to the nearest tenth of a minute, required for the coffee to cool to 30∘C.
(5) Explain the physical significance of both the sign and the magnitude of k in the context of this cooling process.