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Question

Spherical rain drop evaporates ata rate proportional to its surface area.The differential equation correspondingto the rate of change of the radius ofthe rain drop if the constant of proportionality is K>0 is
.

A
drdt+K=0
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B
drdtK=0
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C
drdt+Kr=0
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D
drdtKr=0
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Solution

The correct option is A drdt+K=0
The rate of evaporation is measured by the rate of change of volume ofthe spherical droplet.So if at an instant of time the radius is r Surface area = 4πr2The rate of evaporation=dVdtwhere V=Volume of the dropletdVdt4πr2dVdt=K(4πr2)Here the minus sign is therebecause the the volume is decreasing.V(Volume of sphere)=43πr3dVdt=ddt(43πr3) =43π×3r2drdt =4πr2drdtSo, 4πr2drdt=K( 4πr2) drdt+K=0

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