# 01 - Thermometer reading after 6 minutes of being outside

**Problem 01**

A thermometer which has been at the reading of 70°F inside a house is placed outside where the air temperature is 10°F. Three minutes later it is found that the thermometer reading is 25°F. Find the thermometer reading after 6 minutes.

**Solution 01**

$\dfrac{dT}{dt} = -k(T - T_s)$

$\dfrac{dT}{dt} = -k(T - 10)$

$\dfrac{dT}{T - 10} = -k \, dt$

$\ln (T - 10) = -kt + \ln C$

$\ln (T - 10) = \ln e^{-kt} + \ln C$

$\ln (T - 10) = \ln Ce^{-kt}$

$T - 10 = Ce^{-kt}$

$T = 10 + Ce^{-kt}$

When t = 0, T = 70°F

$70 = 10 + C$

$C = 60$

Hence,

$T = 10 + 60e^{-kt}$

When t = 3 min, T = 25°F

$25 = 10 + 60e^{-3k}$

$15 = 60e^{-3k}$

$\frac{15}{60} = e^{-3k}$

$e^{-k} = (\frac{1}{4})^{1/3}$

Thus,

$T = 10 + 60(\frac{1}{4})^{t/3}$

After 6 minutes, t = 6

$T = 10 + 60(\frac{1}{4})^2$

$T = 13.75^\circ F$ *answer*

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