Question 1096964: Estabon poured himself a hot beverage that had a temperature of 204
degreesF then set it on the kitchen table to cool. The temperature of the kitchen was a constant 69 degreesF. If the drink cooled to 182degreesF in 2 minutes, how long will it take for the drink to cool to 106 degreesF? (Do not round until the final answer. Then round to the nearest minute as needed.)
Answer by jorel1380(3719)   (Show Source):
You can put this solution on YOUR website! According to Newton's law of cooling, the temperature T(t)=T(a)+((T(0)-T(a))e^-kt; where T(t) is the temperature after time t;T(0) is the initial temperature, T(a) is the ambient temperature of the surrounding environment, and k is the cooling constant. So, here we have:
T(2)=182=69+(204-69)e^-2k
Then:
113=135e^-2k
0.83703703703703703703703703703704=e^-2k
ln 0.83703703703703703703703703703704=ln e^-2k=-2k ln e=-2k
k=0.08894347986304444010218177518887
So, in order to reach 106 degrees, we have:
106=69+(204-106)e^-0.08894347986304444010218177518887t
0.37755102040816326530612244897959=e^-0.08894347986304444010218177518887t
ln 0.37755102040816326530612244897959=ln e^-0.08894347986304444010218177518887t=-0.08894347986304444010218177518887t ln e=-0.08894347986304444010218177518887t
t=10.9513 minutes
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