A hot body placed in air is cooled down according to Newton's law of cooling, the rate of decrease of temperature being `k` times the temperature difference from the surrounding. Starting from `t=0` , find the time in which the body will loss half the maximum heat it can lose.

Answer

B

Explanation

Let `T_0` be the temperature of surrounding and T be the temperature of hot body at some instant. Then,
`-(dT)/(dt) = K (T - T_0)`
or `int_(T_m)^T (dT)/(T-T_0) = -K int_(0)^t (dT)`
`(T_m = "temperature at" t= 0)`
Solving this equation, we get
`T = T_0 + (T_m - T_0)e^(-Kt)`......... (i)
Maximum temperature it can lose is `(T_m -T_0)`
From Eq. (i),
`T - T_0 = (T_m -T_0)e^(-Kt)`
Given that
`(T-T_0) = (T_m - T_0)/2 = (T_m -T_0)e^(-Kt)`
Solving this equation we get,
`t = (ln (2))/K` .