A hot body placed in a surrounding of temperature θ 0 obeys Newton's law of cooling dt d θ = − k ( θ − θ 0 ) , where k = 5 l n 10 s − 1 . If the maximum heat that the body can lose is Q 0

Question

A hot body placed in a surrounding of temperature θ 0 ​ obeys Newton's law of cooling dt d θ ​ = − k ( θ − θ 0 ​ ) , where k = 5 l n 10 ​ s − 1 . If the maximum heat that the body can lose is Q 0 ​ . Then find the time (in seconds) starting from t = 0 at which the body will lose 90% of this maximum heat Q 0 ​ .

TestHub · Accepted Answer

Let the initial temperature above surroundings be θ₁ - θ₀.
$Q_{0} = m c (θ_{1} - θ_{0})$
After 90% of the maximum heat has left, 10% of that excess remains.
$\frac{θ ( t ) - θ _{0}}{θ _{1} - θ _{0}} = \frac{1}{10}$
$θ (t) - θ_{0} = (θ_{1} - θ_{0}) e^{- k t}$
$e^{- k t} = \frac{1}{10} \Rightarrow t = \frac{l n 10}{k}$
$k = \frac{l n 10}{5} s^{- 1} \Rightarrow t = 5 s$
Answer 5.

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