An object with $T_0 = x$ is in a room with fixed temp 30 celcius. After ten minutes the objects temp is negative $10$ celsius and after 10 more minutes it is negative $5$ celsius What was the $T_0$? This is a Newton's law of cooling problem.
I guess since there is no k given that the cooling rate is constant and it warms five degrees per 10 minutes to $T_0 = -15$
Best Answer
Here is a road map for you to follow to the problem's solution.
Newton's law of cooling states that the rate at which a body cools is proportional to the difference between its temperature and the surrounding temperature. Assuming a very large room, we have the following differential equation:
$$\frac{dT}{dt}=k(T-30)$$
Now, you want to separate the variables and integrate. This gives:
$$\frac{dT}{T-30}=kdt\quad\Longrightarrow\quad\ln(T-30)=kt+C\quad\Longrightarrow\quad T(t)=Ae^{kt}+30$$
where $A=e^C$. Now, plug in the two ordered pairs you know, $(10,-10)$ and $(20,-5)$, to obtain a system of two equations in the two variables $A$ and $k$.
$$-10=Ae^{10k}+30\quad\Longrightarrow\quad-40=Ae^{10k}\\-5=Ae^{20k}+30\quad\Longrightarrow\quad-35=Ae^{20k}$$
Solve this system to find the values of your constants. To find $k$, divide the two equations:
$$\frac{35}{40}=e^{10k}\quad\Longrightarrow\quad k=\frac{1}{10}\ln\left(\frac{7}{8}\right)$$
Use this to find $A$. Plug $A$ and $k$ back into $T(t)$. Then just plug in $t=0$ to find the initial temperature.