A hot body placed in a surrounding of temperature θ_0 conforms to...

The maximum heat the body can lose is when its temperature drops from $\theta_1$ to the surrounding temperature $\theta_0$.

Maximum heat loss, Q_\max = ms(\theta_1 - \theta_0)

The heat lost by the body at time $t$ when its temperature is $\theta$ is given by:

$Q = ms(\theta_1 - \theta)$

Given that the body loses $90\%$ of the maximum heat it can possibly lose:

Q = 0.9 Q_\max

$ms(\theta_1 - \theta) = 0.9 ms(\theta_1 - \theta_0)$

$\theta_1 - \theta = 0.9(\theta_1 - \theta_0)$

$\theta - \theta_0 = 0.1(\theta_1 - \theta_0)$

From Newton's law of cooling, the temperature $\theta$ at time $t$ is given by:

$\theta - \theta_0 = (\theta_1 - \theta_0)e^{-kt}$

Substituting the value of $\theta - \theta_0$:

$0.1(\theta_1 - \theta_0) = (\theta_1 - \theta_0)e^{-kt}$

$e^{-kt} = \dfrac{1}{10}$

$e^{kt} = 10$

$t = \dfrac{\ln 10}{k}$