Evaporating rain drop! 💧

Calculus Level 3

A spherical raindrop​ evaporates at a rate proportional to its surface area. If its radius is originally 3mm and after one hour it is reduced to 2mm, then which one of the following expressions gives the radius of the rain drop at any instant?

Clarifications : Do not change the units of time and length into SI units.

Try another problem here

ln ⁡ r = − t 2 − 3 \ln r=-t\sqrt{2}-3 ln ⁡ r = − 3 t + 4 \ln r=-3t+4 r = 5 − 7 t r=5-7t r = e − t + 1 r=e^{-t}+1 r = 3 − t r=3-t

Sparsh Sarode by Sparsh Sarode , 22, India

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1 solution

Sparsh Sarode
Jun 18, 2016

Let r be the radius of the rain drop at any instant of time t

It's volume, V = 4 3 π r 2 V=\dfrac{4}{3} \pi r^2 and surface area s = 4 π r 2 s=4 \pi r^2

d V d t = k 4 π r 2 \dfrac{dV}{dt}=k4 \pi r^2 (given) and d V d t = 4 π r 2 d r d t \dfrac{dV}{dt}=4 \pi r^2 \dfrac{dr}{dt}

k 4 π r 2 = 4 π r 2 d r d t k4 \pi r^2=4 \pi r^2\dfrac{dr}{dt}

r = k t + c r=kt+c

When t = 0 , r = 3 t=0, r=3 and when t = 1 , r = 2 t=1, r=2

Thus, k = − 1 , c = 3 k=-1, c=3

∴ r = 3 − t \therefore r=3-t

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Nice!! Gud question

Rishi K - 4 years, 12 months ago

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Thanks..:) :)

Sparsh Sarode - 4 years, 12 months ago

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