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Use differential equations to compute the temperature

by RoRi

A thermometer is stored at 75 degrees. It is taken outside and reads 65 degrees after 5 minutes, and it reads 60 degrees after 10 minutes. Compute the temperature outside.

We use the equations from Newton’s law of cooling (example 3 on page 315 of Apostol). This gives us

    \[ f(t) = be^{-kt} + e^{-kt} \int_a^t k M(u) e^{ku} \, du. \]

We have M(u) = T_0 since the outside temperature is constant, and we are given f(0) = 75 so a = 0, and b = 75. Therefore,

    \begin{align*} f(t) &= 75e^{-kt} + e^{-kt} \int_0^t kT_0 e^{ku} \, du \\[9pt] &= 75 e^{-kt} + e^{-kt} \left( T_0 e^{kt} - T_0 \right) \\[9pt] &= (75-T_0)e^{-kt} + T_0. \end{align*}

We are then given f(5) = 65 and f(10) = 60 so,

    \[ 65 = (75-T_0)e^{-5k} + T_0 \qquad \implies \qquad e^{-5k} = \frac{65-T_0}{75-T_0}. \]

And,

    \begin{align*} &&60 &= (75-T_0) e^{-10k} + T_0 \\[9pt] \implies && 60 &= \frac{(75-T_0)(65-T_0)^2}{(75-T_0)^2} + T_0 \\[9pt] \implies && (60 - T_0)(75 - T_0) &= (65-T_0)^2 \\[9pt] \implies && 4500 - 135T_0 + T_0^2 &= 4425 - 130 T_0 + T_0^2 \\[9pt] \implies && T_0 &= 55. \end{align*}

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