Newton\'s law of cooling is: du/dt= -k(u-T) where u(t) is temperature of an obje
ID: 1948052 • Letter: N
Question
Newton's law of cooling is:du/dt= -k(u-T)
where u(t) is temperature of an object, t is in hours, T is a constant ambient temperature, and k is a positive constant.
Suppose a building loses heat in accordance with Newton's law of cooling. Suppose that the rate constant k has the value 0.15 hr ? 1 . Assume that the interior temperature is Ti = 70?F, when the heating system fails. If the external temperature is T = 11? F, how long will it take for the interior temperature to fall to T1 = 32?F?
Explanation / Answer
du/dt = -k(u - T) du/(u - T) = -kdt integrating both sides ln(u - T) = -kt + c ln(u - T) = -kt + lnC ( writing c = lnC for simplification) (u - T)/C = e^-kt u = T + Ce^-kt Now, T = 11 , k = 0.15 hr^-1 u(0) = T + C = 70 => C = 59 now, u(t) = 11 + 59e^-0.15t = 32 => 59e^-0.15t = 21 e^-0.15t = 21/59 t = -ln(21/59)/0.15 t = 6.8868 hrs