Newton's Law of Cooling

Calculus 1 · Axiom Academy

EXAMPLE Newton's Law of Cooling Solving an exponential-decay temperature model to predict a future temperature. A cup of coffee at is set down in a room held at . After 5 minutes it has cooled to . Using Newton's Law of Cooling , find the coffee's temperature after 10 minutes. The temperature falls fast at first, then flattens toward the room temperature — never quite reaching it. We know two points on this curve; the model lets us read off the rest. Nicely done. You turned two data points into a full cooling model and used it to predict a temperature you were never told. Solution form: Newton's Law solves to T(t) = T_s + (T_0 - T_s)e^ -kt — the gap above the surroundings decays exponentially. One unknown, one data point: the second measurement T(5) = 150 is exactly what you need to pin down the rate . Result: — note it dropped in the first 5 minutes but only about in the next 5, because cooling slows as the gap shrinks. The same exponential-decay model fits anything relaxing toward equilibrium — a cooling engine, a charging capacitor, a drug clearing the bloodstream.

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