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Question

Assuming Newton's law of cooling to be valid, the time taken by a body to cool from a temperature T1 to a temperature T2 in a room where the temperature is T0 (T0<T2<T1) is proportional to


  1. loge(T1T0T2T0)

  2. loge(T2T0T1+T0)

  3. loge(T1T2T0)

  4. loge(T1+T0T2+T0)


Solution

The correct option is A

loge(T1T0T2T0)


According to Newton's law of cooling, the rate of fall of temperature is given by
dTdt=K(TT0)
where K is a constant, T is the temperature of the body at time t and T0 that of its surrounding.
Hence, Kdt=dTTT0
Integrating, we have
Kt0dt=T2T1dTTT0Kt=|loge(TT0)|T2T1=loge(T2T0T1T0)=loge(T1T0T2T0)t=1Kloge(T1T0T2T0)
Hence, the correct choice is (a).

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