Athermometer is taken from an inside room to the outside, where the air temperature is 30â° f. after 1 minute the thermometer reads 80â° f, and after 5 minutes it reads 55â° f. what is the initial temperature of the inside room? (round your answer to two decimal places.)

We can solve the problem by using Newton's law of cooling, which gives the rate of temperature variation of the thermometer given the temperature of the air ()

Where T is the temperature in Fahrenheit and t is the time in minutes.
We also know the temperature after t=1 min and t=5 min:
(A) T(1)=80
(B) T(5)=55
The problem asks to find the temperature at time t=0. 

Let's solve the differential equation. We can rewrite it as:

by integrating on both sides, we have



where we called .

Nowe we have to find k and C. We can use conditions (A) and (B) for this purpose. We have the following system of equations:
(A)
(B) 
From which we find



And finally, we can substituting these values into the main expression of T, and calculating it at time t=0:

and using t=0, we find

which is the initial temperature of the thermometer.