Question

A thermometer is taken from a room where the temperature is 20°C to the outdoors, where the temperature is 2°C. After one minute the thermometer reads 11°C.temperature after one more minute ?

Answer 1

This deals with Newton's Law of Cooling. According to his law, the rate of change of the temperature is proportional to the difference between thetemperature of the thermometer (at some time) and the ambient temperature. In other words, the equation we can use for this problem is derived from thefollowing differential equation, where T is the temperature, T_ais the ambient temperature (and could, technically, depend on time),and k is some constant:

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In turns out that the solution to this differential involves Euler's number and the exp(x) function. It is given by (where T₀is theinitial temperature):

We can use this equation, but we need to first determine the value of the constant k for our specific situation. Hence, we substitute the values givenfor a temperature at a certain time t = 1 min (notice we can use any time standard as long as we are consistent because they are all related byconstants), and solve for k:

Now, to find the temperature at t = 2 min (one minute later), we simply resubstitute, using our value determined for k (k = ln2):

Hence, the temperature after one more minute passes is T(2) = 13/2 = 6.5 ^oC.