heating and cooling newtons law

Question

Newton's Law of Heating and Cooling gives the model below. For water, k = -0.04095. Assume that you are in a 76 °F room with a cup of water. The water temperature is 34.3 °F. Find the water temperature in the cup after 10 minutes.

 

t=a+se^kt

Answer 1

The water ends up at about 48.3 °F.

Before the arithmetic, notice something about this problem: the water is at 34.3 °F in a 76 °F room, so it is colder than its surroundings. It's going to warm up, not cool down. That's why the law is called heating and cooling — one formula covers both directions, and nothing about it needs to change.

The model

T(t) = T_room + (T₀ − T_room)e^kt

(If your textbook writes it slightly differently, check it matches this before using my numbers.) Read it as: the room temperature, plus whatever is left of the original gap. The factor e^kt is the fraction of the gap that survives after t minutes.

Substitute

T_room = 76, T₀ = 34.3, k = −0.04095, t = 10.

The starting gap is T₀ − T_room = 34.3 − 76 = −41.7. Keep that negative sign — it is what tells the formula to warm the water instead of cooling it.

e^{(−0.04095)(10)} = e^−0.4095 ≈ 0.6640

T(10) = 76 + (−41.7)(0.6640) = 76 − 27.69 = 48.31 °F

Sanity-check it

Three quick checks, and they're worth doing every time:

Is it between the start and the room? 34.3 < 48.3 < 76. Yes. The water can approach room temperature but can never pass it, so an answer above 76 or below 34.3 would mean an algebra slip.

Does t = 0 give back the start? e⁰ = 1, so T(0) = 76 + (−41.7)(1) = 34.3. Correct.

Does the direction make sense? It warmed from 34.3 toward 76. Yes.

The idea underneath

After 10 minutes, e^−0.4095 ≈ 0.664 means about 66% of the original gap is still there — the water closed only about a third of the distance to room temperature. The gap shrinks by the same proportion in equal time intervals, not by the same number of degrees, which is what makes this exponential rather than linear.

You can put a number on that: the gap halves every ln2 / 0.04095 ≈ 17 minutes, no matter how large it is. So at 17 minutes the gap is about 21 degrees, at 34 minutes about 10 degrees, and so on. The water gets very close to 76 °F but technically never arrives — 76 is a horizontal asymptote.

One last note on the sign of k: it must be negative for this model to make physical sense. A negative k makes e^kt decay toward zero, which shrinks the gap. A positive k would make the gap grow forever, meaning a cup of water spontaneously running away from room temperature.