Question

A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. if a sheet hung in the wind loses half its moisture during the first hour, then the time when it would have lost $99.9\%$ of its moisture is (weather conditions remaining same)

• A. Less than $5$ hours
• B. More than $10$ hours
• C. Approximately $10$ hours
• D. Approximately $9$ hours

Answer 1

Answer: (A)

Less than $5$ hours

$\begin{array}{c}Given,\\ \frac{dC}{dt}=aCwherea=cons\tan t\\ c_1=Initialmoisture\\ {\int }_{c_1}^{\frac{c_1}{2}}\frac{dC}{C}={\int }_0^{1}adt\\ {\left(lnC\right)}_{c_1}^{\frac{c_1}{2}}={\left(at\right)}_0^1\\ \Rightarrow a=-\ln 2\\ {\int }_{c_1}^{.05c_1}\frac{dC}{C}={\int }_0^t-\ln 2.dt\\ \Rightarrow t=\frac{\ln 20}{\ln 2}hr\end{array}$