Question

According to Newton's law of cooling, the temperature of a body is \( T \) at time \( t \) and the temperature of the surroundings is \( T_s \). The rate of change of temperature is given by \[ \frac{dT}{dt} = k(T_s - T) \] If the initial temperature at time \( t = 0 \) is \( T_0 \), then the temperature \( T \) is:

• A. \( T_s - (T_s - T_0) e^{-kt} \)
• B. \( (T_s - T_0) e^{-kt} + T_0 \)
• C. \( (T_0 - T_s) e^{-kt} + T_s \)
• D. \( T_0 + (T_0 - T_s) e^{-kt} \)

Hint:

Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of the body and the surroundings.

Answer 1

The Correct Option is C

The differential equation for Newton's law of cooling is: \[ \frac{dT}{dt} = k(T_s - T) \] The solution to this differential equation is: \[ T(t) = T_s - (T_s - T_0) e^{-kt} \] Thus, the temperature of the body at time \( t \) is \( T(t) = T_s - (T_s - T_0) e^{-kt} \).