Question

Newton's law of cooling is dudt=−k(u−T), where u(t) is the temperature of an object, t is in hours, T is a constant ambient temperature, and k is a positive constant. Suppose a building loses heat in accordance with Newton's law of cooling. Suppose that the rate constant k has the value 0.17 hr-1. Assume that the interior temperature is Ti=77°F, when the heating system fails.

Required:
If the external temperature is T=10°F, how long will it take for the interior temperature to fall to T1=35°F?

Answer 1

5.80 hours

Explanation:

The temperature difference is decaying from (77 -10) = 67 to (35 -10) = 25. The time required for that to happen satisfies ...

25 = 67e^(-kt)

ln(25/67) = -kt . . . . . divide by 67, take natural logs

t = ln(25/67)/(-0.17) ≈ 5.80 . . . hours