Homework SourseNewtons Law of Cooling states that the rate at which an obje
Newton\'s Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and the surrounding medium. Thus, if an object is taken from an oven at 309 F and left to cool in a room at 75 F, its temperature T after t hours will satisfy the differential equation dT/dt=k(T-75). If the temperature fell to 201 F in 0.6 hour(s), what will the temperature be after 5 hours?
Hint: Separate variables.
Hint: Separate variables.
Solution
dT / dt = k(T - 75) dT / (T - 75) = k dt ln (T - 75) = kt + c convert to exponential: e^(kt + c) = T - 75 C e^(kt) = T - 75 T = 75 + Ce^(kt) we know that T(0) = 309 so 309 = 75 + C(1) C = 309 - 75 = 234 T = 75 + 234 e^(kt) another data point: T(.6) = 201 sub to find k: 201 = 75 + 234 e^(.6k) 126 = 234 e^(.6k) 126 / 234 = e^(.6k) ln (126 / 234) = .6k ln (126 / 234) / .6 = k now we know that T(t) = 75 + 234 e^(t ln (126 / 234) / .6) plug in t = 5 to find T(5), or the temperature after 5 hours T(5)= 76.34ºF
