Use this interactive figure to explore Newton's law of cooling for different initial temperatures, room temperatures, and rates of cooling.[br][br]Newton's law of cooling says that if [math]T[/math] is the temperature of the object at time [math]t[/math] and [math]T_s[/math] is the constant surrounding temperature, then the rate of change of the temperature of the object is [center][math]\frac{dT}{dt}=-k(T-T_s).[/math][/center]We might call [math]k[/math] the [i]proportional [/i]rate of cooling; it is the rate of cooling per degree difference between the object and its surrounding. Solving the differential equation yields[br][center][math]T=T_s+(T_0-T_s)e^{-kt},[/math][/center]where [math]T_0[/math] is the temperature of the object at [math]t=0[/math].[br][br]In this interactive figure you can control the intial temperature of the object [math]T_0[/math], the surrounding temperature [math]T_s[/math], and the proportional rate of cooling [math]k[/math] to see their effect on the temperature of an object over time.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]