Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 2258404 • Letter: N
Question
Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings.
Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. Let k>0 be the constant of proportionality. Assume the coffee has a temperature of 210 degrees Fahrenheit when freshly poured, and 2 minutes later has cooled to 191 degrees in a room at 70 degrees.
a) Write an initial value problem for the temperature T of the coffee, in Fahrenheit, at time t in minutes. Your answer will contain the uknown constant k
b)Solve the initial value problem in part (a). Your answer will contain the unknown constant k.
c)Determine the value of the constant k
d)Determine when the coffee reaches a temperature of 151 degrees.
Explanation / Answer
Solving all 4 parts
T(t) = c*e-k(t) +T0
Given, T0 = 70, Initially time,t =0, T(t=0) = 210
Thus, 210 = c*e0 + 70 => c= 140
Thus, T(t) = 140e-k(t) + 70
putting T(t=2) = 191 and t = 2 we get
191 = 140*e-2k + 70
Solving, we get
e-2k = (191-70)/140 = 0.864
Thus,-2k logee = loge(0.864)
-2K = -0.146
Thus, K = 0.073
T(t) = 140e-0.073t + 70
When,T(t) = 151
We get, 151 = 140e-0.073t + 70
Thus, e-0.073t = (151-70)/140 = 0.579
Thus -0.073t = ln(0.579)
Thus, t = 7.5 mins = 7 min 30 sec