Problem 2. Newton's law of cooling states that the rate of change with respect to time (let's say per hour) of the temperature T of a "body" is proportional to the difference between T and the ambient temperature of the surrounding medium. More precisely, if T (t) denotes the temperature of a (glass) bottle of Topo Chico at a time t, then its temperature changes per hour according to the differential equation ( ) T = r(T A), where r is a positive constant and A is the ambient temperature in, like, you know, my office (PMA 13.150), which we can also regard as a constant (because the thermostat in my office is purely decorative). (a) Assume I pull a Topo from the fridge in the 12th-floor PMA lounge at 45 F, and set it on one of the three desks I claim as my own in my office, in which the ambient temperature is 65 F. Use the formula given in class to write down the solution to the initial value problem consisting of the equation () and the initial value T (0) = 45. (Your solution should have an r in it.) (b) One hour after placing the Topo on my desk, its temperature is 50 F. Use this information and your solution for (a) to determine the value of the relative growth rate r. Show all of your work. 2

(c) How many hours will it take for the bottle of Topo to reach a temperature of 60 F? Show all of your work.