Newton\'s law of cooling states that the temperature of an object changes at a r
ID: 2973691 • 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. If the coffee has a temperature of 185 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 175 degrees in a room at 78 degrees, determine when the coffee reaches a temperature of 150 degrees. The coffee will reach a temperature of 150 degrees in__________ minutes.Explanation / Answer
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. If the coffee has a temperature of 195 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 185 degrees in a room at 66 degrees, determine when the coffee reaches a temperature of 145 degrees. The coffee will reach a temperature of 145 degrees in __________ minutes. dT/dt = -k(T-S) where T is current temperature and S = ambient temperature dT/(T-S) = -k.dt Solving the differential equation gives ln(T-S) = -kt + C T-S = e^(-kt+C) T(t) = S + e^(-kt+C) T(t) = S +(To-S)*e(-kt) where To = initial temperature at t = 0 T(t) = 66+(195-66)*e^(-kt) T(t) = 66+129^(-kt) 185=66+129*e^-2.5k 119=129*e^(-2.5k) ln(119/129) = -2.5k k = 0.0323 T(t) = 66 + 129*e^(-0.0323t) 145 = 66 + 129*e^-0.0323t) 79/129 = e^(-0.0323t) ln(79/129) = -0.0323t t = 15.2 minutes http://www.endmemo.com/physics/coollaw.php