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?

Question

Dalton Phillips

Mathematics

30 October, 04:56

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?

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Fellow · Answer 1

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