mathematically model how a coffee cup’s temperature varies over time.

Question

It has been proposed that the ideal temperature for producing a coffee is 900C. However, this is too hot to enjoy the beverage.

Consider a hot cup of coffee. The temperature in the coffee cup will decrease at a rate proportional to the difference in the current temperature and the ambient temperature. Based on this statement, show how we can model our coffee cup with the ODE

dT/dt + kT = kT_Infinity

Answer 1

dT(t)/dt = k * (T-T_amb), where T(t) is the temperature of the coffee at time t, and T_amb is the ambient temperature, which is generally presumed to be constant, though it could be a function of time also.

"decrease at a rate" means derivative of the value of T.

"proportional to the difference" means a constant times the difference. k is the constant of proportionality in the above ODE.