Athermometer is removed from a room where the temperature is 70 f and is taken outside where the air temperature is 10 f. after one half minute the thermometer reads 50 f. what is the reading of the thermometer at t=1min? how long will it take for the thermometer to reach 15 f?

This question is the application of differential eqns in order to derive a model for the temperature dependence with time. Actually, a general equation has already been derived for this type of cases. This equation is known as the Newton's Law of Cooling. The equation is 

(T - Ts) / (To -Ts) = e^(-kt)

where T is the the temperature at any time t
Ts is the surrounding temperature
To is the initial temperature
k is the constant
t is the time

several assumptions have been made to arrive at this form, i suggest you trace the derivation of the general formula.

First we need to look for k using the initial conditions that is @t = 1.5 min, T = 50 F

substituting we get a k = 0.2703

therefore @ t = 1 min, T = 55.79 F

@ T = 15 F the time required is 9.193 min.

 

The answer would be 13.5 minutes.