Question

b) If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content and if it loses half of its moisture during first 10 minutes, when will it have lost 99% of its moisture?​

Answer 1

Given:

If a wet sheet in a dryer loses its moisture at a rate proportional to its moisture content and if it loses half of its moisture during the first 10 minutes,

To Find:

when will it have lost 99% of its moisture?​

Solution:

It is given that dryer loses its moisture at a rate proportional to its moisture content, let the moisture at any time t be M, so the dryer will lose moisture

$\frac{-dM}{dt} \propto M$

Now losing the proportionality sign we will get a constant,

$\frac{-dM}{dt} =kM$

Now integrating the equation taking Mo at t=0, we get

$\frac{-dM}{dt} =kM\\\frac{dM}{M} =-k dt\\\int\limits^M_{M_{0}} {\frac{1}{M} } \, dM=\int\limits^t_0 {-k} \, dt\\log_e \frac{M}{M_0}=-kt\\M=M_oe^{-kt}$

Now finding the value of k we have,

$\frac{M_o}{2} =M_oe^{-k*10}\\k=\frac{1}{5}$

Now, when it loses 99% of its moisture, so t will be

$\frac{1}{100}=e^{-\frac{t}{5}}\\t=23.03$

Hence, it will lose 99% of its moisture in 23.03 minutes.