differential equation

Question

A thermometer is removed from a room where the temperature is 70° F and is taken outside, where the air temperature is 20° F. After one-half minute the thermometer reads 50° F. What is the reading of the thermometer at t = 1 min?

 (Round your answer to two decimal places.)

_______________ ° F

How long will it take for the thermometer to reach 25° F? (Round your answer to two decimal places.)

 _______________min

Answer 1

Let T(t) = temperature reading on the thermometer at time t minutes after leaving the room

By Newton's Law of Cooling, dT/dt = k(T - 20)

dT / (T - 20) = kdt

∫[dT / (T - 20) = ∫kdt

ln l T - 20 l = kt + C

T - 20 = ±e^kt+C

T - 20 = ±e^Ce^kt = Ae^kt, where A = ±e^C

So, T = Ae^kt + 20

T(0) = 70, so A + 20 = 70. Therefore, A = 50.

T(t) = 50e^kt + 20

Since T(0.5) = 50, we have 50e^0.5k + 20 = 50;

e^0.5k = 0.6

0.5k = ln(0.6) So, k = -1.02165.

T(t) = 50e^-1.02165t + 20

T(1) = 50e^-1.02165 + 20 = 38.00°

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T(t) = 50e^-1.02165t + 20 = 25

Solve for t.