A hot body placed in a surrounding of temperature `theta_(0)` obeys Newton's law of cooling `(d theta)/(dt)=-k(theta-theta_(0))` . Its temperature at `t=0` is `theta_(1)` the specific heat capacity of the body is `s` and its mass is `m`. Find
(a) the maximum heat that the body can lose and
(b) the time starting from `t=0` in which it will lose `90%` of this maximum heat.

Answer

(a) `ms(T_(1)-T_(0))` (b) `(In2)/(k)`

Explanation

(a) `Q_(max) = ms (T_(1)-T_(0))`
The body will kept losing heat unitl it comes in thermal equilibrium with the surrounding.
(b) `0.5 ms (T_(1)-T_(0)) = ms (T_(1)-T)`
`rArr T= 0.5 (T_(1)+T_(0))`
and `(dT)/(dt) = -k (T - T_(0))`
`overset(0.5(T_(1)+T_(0)))underset(T_(1))int(dT)/(-(T-T_(0)))=overset(t)underset(0)intkdt or t = (ln2)/(k)`