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October 24th, 2007, 09:38 PM   #1
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Confusing problem

Hey everyone, this is a problem I have been assigned and I am stuck on it.

Physics tells us that a drop of mist light enough to float in the air and not fall to the ground as a raindrop is shaped like a perfect sphere, and water molecules can join this drop from any direction from outside the drop itself- meaning that the drop gains moisture at a rate proportional to its surface area. Show that, under these conditions, the radius of the drop is growing at a constant rate, no matter how large the drop currently is.

I know that I need to use the formula for the surface area of a sphere and possibly its derivative, but I just am confuesed as where to go next. Any help is greatly appreciated.
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October 24th, 2007, 10:02 PM   #2
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try this

water molecules can join this drop from any direction from outside the drop itself- meaning that the drop gains moisture at a rate proportional to its surface area.

this translates to:
dV = k A
dV = k 4 (pi) r^2
where dV is the change in volume, k is a constant, and A is the surface area.

The formula for the volume of a sphere is this:
V = (4/3)(pi)r^3

Now, taking the derivative of the volume of a sphere, we can write
dV = 4(pi)r^2 dr

Substitute dV from above, and we have
k 4 (pi) r^2 = 4(pi)r^2 dr

Cancel 4 (pi) r^2 from both sides, and we have
k = dr

Therefore, the change in radius is constant
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October 24th, 2007, 10:33 PM   #3
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Ok I don't understand the need for the constant k.

I i think i was initially thinking to hard about the problem, this is what I came up with.

Surface Area of a sphere- a= 4(pi)r^2

So the change in the surface area = da/dt = 8(pi)r dr/dt

so the change in the radius dr/dt = 1/8(pi)r x da/dt

I believe this is what i'm supposed to show, what do you guys think?
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October 24th, 2007, 11:53 PM   #4
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i think the problem with leaving it as dr/dt = 1/8(pi)r x da/dt is that according to this equation, the change in the radius is dependent on r and the rate of change in the area. they want to see that the rate of change is constant.
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October 25th, 2007, 12:19 AM   #5
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ok that makes sense, but where does the constant k come from, what is it, what does it stand for, im just confused as to where you got the k from.
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October 25th, 2007, 01:43 AM   #6
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i originally had dV = kA because the question read, "drop gains moisture at a rate proportional to its surface area"

i translated this to mean, "the volume of the drop changes directly in proportion to the surface area", making this a "direct variation" equation (y=kx)

i used "k" as the constant of proportionality.
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