Newton\'s law of cooling states that the temperature of an object changes at a r

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 195 degrees Fahrenheit when freshly poured, and 1.5 minutes later has cooled to 177 degrees in a room at 64 degrees, determine when the coffee reaches a temperature of 137 degrees. The coffee will reach a temperature of 137 degrees in?

Explanation / Answer

Newton's law of cooling can be stated mathematically as follows,

dT/dt = -k(T-S) where T is current temperature and S = Room temperature

dT/(T-S) = -k.dt

Solving the differential equation with integrating both the sides, gives

ln(T-S) = -kt + C

T-S = e^(-kt+C)

T(t) = S + e^(-kt+C)

T(t) = S +(T0-S)*e(-kt) where T0 = initial temperature at t = 0

T(t) = 64+(195-64)*e^(-kt)

T(t) = 64+131*e^(-kt)

177 = 64+131*e^(-1.5k)

113 = 131*e^(-1.5k)

113/131 = e^(-1.5k)

Taking natural logarithm at both sides, we have

ln(113/131) = -1.5k

k = 0.0985

T(t) = 64 + 131*e^(-0.0985t)

137 = 64 + 131*e^-0.0985t)

73/131 = e^(-0.0985t)

Again taking natural logarithm at both sides, we have

ln(73/131) = -0.0985t

t = 5.94 minutes, approx. 6 minutes

Hence, the coffee will reach at the temperature of 137 degrees at approx. 6 mins.

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