Answer to Question #259205 in Differential Equations for Swapnil

Question

1. 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 min of drying, when will it be practically dry, say, when will it have lost 99% of its moisture?

Answer 1

Let the moisture content present in the sheet at any time t be M.

dryer loses moisture content at $\dfrac{-dM}{dt}$

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

(k = proportionality constant)

$\dfrac{dM}M = - k dt$

Integrating,

$\log_e^{\dfrac M{M_0}}=-kt$

(initial moisture content at t = 0 be M₀)

$M = M_o e^{-kt}$ ⇒ (1)

First 10 mins, t = 10, $M =\dfrac {M_{0}}{2}$

$\dfrac{M_{0}}2 = M_{0} ×e ^{ (- k ×10)}\\ k = \dfrac{\log e ^ 2}{10}=\dfrac 15$ ⇒ (2)

Merging, equations 1 and 2,

$\dfrac{M}{M_0}=e^{-\frac t5}⇒\dfrac{1}{100} = e^{-\frac t5}$

$t =23.03\textsf{ minutes}$