Newton's Law of Cooling Relates Temperature to Time In this course we start with the solution to Newton's mathematical equation that is solved using calculus. We can use the solution, though, to learn much about solving exponential equations. Here is the solution from calculus: T(t) = Ts+ (To T)e where T} is the initial temperature (called an initial condition in a course in differential equations), T is the temperature of the surroundings, assumed to be constant, T'(t) is the temperature function of time, t, and k is the decay constant. In this version of the equation, k is a positive constant that is determined by the data from a specific set of observations that you will measure. Another team might have a different set of observations due to a different initial temperature of their cup of water, etc. All possible values of k, however, should be similar in magnitude for a given classroom, since the hot water temperature and the air temperature of the room are approximately the same for each team performing the experiment. Part 2 - Finding Function from Three Temperature Readings Example: Suppose the initial temperature of the water is 180 F (Fahrenheit), the air temperature in the room is 72, and the measured temperature after 5 minutes is 140. Thus, we have: Initial Temperature of Water: T, = F Surrounding Temperature: T, = F Temperature after 5 minutes: T(5) = F We need to find the value of k, the decay constant. Finding k: T()= T.+ (TyT)e Substitute T,, and T, into the equation: Enter t=5 into your equation and change T(5) to 140 Simplify the numbers in parentheses: Isolate the term with k on one side: (i) Subtract 72 from both sides: (ii) Divide both sides by 108 (retain fraction form): (iii) Take the natural log of both sides: Recall: Ine = 1 (iv) Use relevant properties to solve for k: (v) Express using 4 decimal place values Mote: k should be positive =k