Geometrical Applications of Differential Equations
While doing a...
Question
While doing an experiment in chemistry lab, Komal found that a substance loses its moisture at a rate proportional to the moisture content. If the same substance loses half of its moisture during the first hour, when will it lose 99% ?
A
ln100ln2
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B
In 50
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C
ln50ln2
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D
In 25
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Solution
The correct option is Aln100ln2 We want to help Komal in finding when the substance will lose 99% moisture. We are given that the rate at which the moisture is escaping/evaporates is proportional to the moisture present in the substance. Let’s represent amount of moisture by m. Rate at which the moisture is escaping/evaporates is proportional to the moisture ⇒dmdt=km, where k is a constant We will solve this differential equation to get a relation between m and time t. ⇒dmm=kdt ⇒lnm=kt+c Let us assume that the amount of moisture at t = 0 is M. ⇒lnM=0+c Or c = ln M ⇒lnm=kt+lnM ⇒lnmM=kt−−−−−−−−−−−−−−−−−(1) We are given one more information that the substance loses half of its moisture in one hour. We will use this to find the value of k. t=1hour⇒m=M2 ⇒ln(M2M)=k×1 ⇒k=−In2 We want to find the time at which 99% of moisture is lost. If 99% of moisture is lost, then 1% of moisture will be remaining ⇒m=M100 Substituting in (1) , we get ln(M100M)=kt=(−ln2)t ⇒−ln100=(−ln2)t ⇒t=ln100ln2