Question

A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is 152 degrees Fahrenheit. After another minute its temperature is 112 degrees Fahrenheit. Find the outside temperature.

Answer 1

@ganeshie8 @hartnn @wio @perl @dan815 @TuringTest @Compassionate

Answer 2

@Directrix @CGGURUMANJUNATH

Answer 3

If the outside temperature is \(T_{out}\), then we have some equation the form of: \[
T(t) = (T_{0}-T_{out})e^{kt}+T_{out}
\]

Answer 4

We can say \(T_0\) is the initial temperature... the temperature of boiling water.

Answer 5

Which is \(212^\circ F\), so:\[
T(t) = (212^\circ F-T_{out})e^{kt}+T_{out}
\]So now we have two variables here... \(k\) and \(T_{out}\). We also have two equations. \[
T(1) = 152^{\circ}F = (212^\circ F-T_{out})e^{k}+T_{out}\\
T(2) = 112^{\circ}F = (212^\circ F-T_{out})e^{2k}+T_{out}
\]

Answer 6

Subtracting \(T_{out}\) from both sides of both equations and dividing the second equation by the first equation gives: \[
\frac{T_{out}-112^{\circ}F}{T_{out}-152^\circ F} = e^{k}
\]

Answer 7

Substituting that into the first equation gives:\[
152^{\circ}F = (212^\circ F-T_{out})\left(\frac{T_{out}-112^{\circ}F}{T_{out}-152^\circ F} \right)+T_{out}
\]

Answer 8

How did you get 212 degrees Fahrenheit?

Answer 9

Boiling water

Answer 10

Thanks!