1. ## [SOLVED] Exponential models

The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.

2. Originally Posted by friend37866
The temperature in a room is 21 degrees Celsius. A thermometer which has been kept in it is placed outside. After 5 minutes the thermometer reading is 16 Celsius. Five minutes later, it is 13 Celsius. Find the outside temperature.
This is related to Newton's Law of Cooling, I presume. (See part way down the page of this link.)

We have that
$T(t) = T_{env} + (T(0) - T_{env})e^{-t/t_0}$
where $T(0)$ is the temperature of the body (in this case the thermometer), $T_{env}$ is the temperature of the environment, and $t_0$ is the "time constant," which we don't know yet.

We know that $T(0) = 21~^oC$ and that at $t = 5~min$, $T(5) = 16~^oC$, and at $t = 10~min$, $T(10) = 13~^oC$.

So:
$T(5) = 16 = T_{env} + (21 - T_{env})e^{-5/t_0}$
and
$T(10) = 13 = T_{env} + (21 - T_{env})e^{-10/t_0}$

Two equations and two unknowns. We want $T_{env}$ and don't care about the time constant, so what I recommend you do is, instead of trying to solve for $t_0$ directly, solve the top equation for $e^{-5/t_0}$. Then note that $e^{-10/t_0} = \left ( e^{-5/t_0} \right )^2$ and use that for your substitution. You'll be left with an equation for $T_{env}$.

Sorry, I gotta run.

-Dan