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Newton’s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. So the rate of cooling for a bottle of lemonade at a room temperature of 75°F which is placed into a refrigerator with temperature of 38°F can be modeled by dT/dt=k(T-38) where T(t) is the temperature of the lemonade after t minutes and T(0) = 75. After 30 minutes the lemonade has cooled to 60°F, so T(30) = 60.

To the nearest degree, what is the temperature of the lemonade after an additional 30 minutes?

Respuesta :

diene
dT/dt=k(T-38)
ln(T-38) = kt+C
T=38 + Ce^(-kt)
Since T(0)=75, 75 = 38+C, so C=37
Since T(30)=60, 60 = 38+37e^(-30k), so k = -ln(22/37)/30 ~= -0.017329
T(60) = 38+37e^(-0.017329*60) = 51.1°F

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