Mr. Marks discovers that the temperature of freshly brewed coffee at local coffe

Question

Mr. Marks discovers that the temperature of freshly brewed coffee at local coffee shop is 130degreeF. He warns the company that if a customer spills hot coffee on them, a lawsuit might result. Assuming that room temperature is a constant 72degreeF, and the temperature of coffee cools to 120degreeF after 4.3 minutes. The company determines that it is safer to serve the coffee at a temperature of 105degreeF. How long does it take the coffee to cool to this temperature? Write the equation that models this scenario.

Explanation / Answer

This is a problm of newton's law of cooling

T(t) = TA + (TH-TA) e^-kt

where
T(t) = Temperature at time t
TA = Ambient temperature (temp of surroundings)
TH = Temperature of hot object at time 0
k = positive constant
t = time

In our case

T(t) = 72 + (130-72) e^-kt=72+58.e^-kt

Now at t=4.3 mins

120=72+58e^-4.3k

e^-4.3k=48/58

k=ln(58/48)/4.3=.044

105=72+58e^-.044t

t=ln(58/33)/.044=12.82mins

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