A body cools from a temperature $3T$ to $2T$ in $10$ minutes. The room temperature is $T.$ Assume that Newton's law of cooling is applicable. The temperature of the body at the end of next $10$ minutes will be
NEET 2016, Diffcult
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According to $Newton's$ law of cooling,
$\frac{{dT}}{{dt}} = K\left( {T - {T_s}} \right)$
For two cases,
$\frac{{d{T_1}}}{{dt}} = K\left( {{T_1} - {T_s}} \right)\,and\,\frac{{d{T_2}}}{{dt}} = K\left( {{T_2} - {T_s}} \right)$
$Here,\,{T_s} = T,\,{T_1} = \frac{{3T + 2T}}{2} = 2.5\,T$
$and\frac{{d{T_1}}}{{dt}} = \frac{{3T -2T}}{{10}} = \frac{T}{{10}}$
${T_2} = \frac{{2T + T'}}{{2}}and\frac{{d{T_2}}}{{dt}} = \frac{{2T - T'}}{{10}}$
$So,\,\frac{T}{{10}} = K\left( {2.5\,T - T} \right)$ $...(i)$
$\frac{{2T - T'}}{{10}} = K\left( {\frac{{2T + T'}}{2} - T} \right)$ $...(ii)$
Dividing eqn. $(i)$ by eqn. $(ii)$, we get
$\frac{T}{{2T - T'}} = \frac{{\left( {2.5T - T} \right)}}{{\left( {\frac{{2T + T'}}{2} - T} \right)}}$
$\frac{{2T + T'}}{2} - T = \left( {2T - T'} \right) \times \frac{3}{2}$
$T' = 3\left( {2T - T'} \right)\,\,or,\,\,4T' = 6T\,\,\therefore \,\,T' = \frac{3}{2}T$
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