So the question reads as following:
A bowl of soup was brought into a room. The soup’s temperature was initially 120 degrees. After 10 minutes, its temperature was 105 degrees. After another 10 minutes, its temperature was 95 degrees.
Use Newton’s Law of Cooling to estimate the temperature of the room. Round to the newest integer.
So I tried to get help from someone I know and they told me to use the second and third points (10,105 and 20,95) to work out the relationship between k and C to use that to find the relationship between C and $T_a$. So I started out by setting both the second and third points equal to $T_a$, then set them equal to each other:
$105-C \cdot e^{-10k}=T_a$ and $95-C \cdot e^{-20k}=T_a$, $105-C \cdot e^{-10k}=95-C \cdot e^{-20k}$
But, at this point I was getting kind of lost.
Honestly I don't really know where to go from here and would appreciate any help in figuring it out.
Best Answer
The formula you want to use can be written as $$T(t)-T_R=(T(0)-T_R) e^{-kt}$$ Applying after 10 minutes, you get $$105-T_R=(120-T_R)e^{-10k}$$ Here $T_R$ is the room temperature. After 20 minutes you have $$95-T_R=(120-T_R)e^{-20k}$$ We want to eliminate the exponential term. You have two equivalent options. Square the first equation, then divide the first equation by the second. The other option is to write $$(120-T_R)e^{-20k}=(120-T_R)e^{-10k}e^{-10k}=(105-T_R)e^{-10k}$$ Once again you divide, and you obtain $$(105-T_R)^2=(120-T_R)(95-T_R)$$ You can see that $T_R^2$ terms will cancel, and you have a simple linear equation.