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A piece of metal is heated to a temperature theta and then left to cool in a room maintained at temperature theta0 (with theta0 less than theta). Which of the following best describes the graph of the metal's temperature T versus time t?

1. A curve that starts at theta, falls steeply at first and then more gradually, approaching the room temperature theta0 asymptotically (exponential decay).
2. A straight line falling at a constant rate from theta down to theta0.
3. A curve that falls slowly at first and then more and more steeply toward theta0.
4. A horizontal straight line that stays constant at theta for all time.

Correct answer: A curve that starts at theta, falls steeply at first and then more gradually, approaching the room temperature theta0 asymptotically (exponential decay).

Solution

By Newton's law of cooling, dT/dt = -k(T - theta0), whose solution is T = theta0 + (theta - theta0) e^(-k t). This is an exponential decay from theta toward theta0: steepest near t = 0 (large T - theta0) and flattening as it approaches the asymptote theta0. It is not linear and never falls below theta0.

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