Evaporating Mothball

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turophile

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Here's the problem:

As a spherical mothball evaporates, its volume decreases at a rate proportional to its surface area. Show that the rate of decrease of the radius is constant.

Here's what I've done so far:

Let r be the radius of the sphere, V its volume, and A its surface area. We know V = (4/3)?r[sup:3pja7b27]3[/sup:3pja7b27] and A = 4?r[sup:3pja7b27]2[/sup:3pja7b27]. We are given dV/dt = c(dA/dt), where c is a nonzero constant. We want to show that dr/dt is a constant.

V = (4/3)?r[sup:3pja7b27]3[/sup:3pja7b27] ? dV/dt = (4/3)? ? d(r[sup:3pja7b27]3[/sup:3pja7b27])/dt = (4/3)? ? 3r[sup:3pja7b27]2[/sup:3pja7b27] ? dr/dt

A = 4?r[sup:3pja7b27]2[/sup:3pja7b27] ? dA/dt = 4? ? d(r[sup:3pja7b27]2[/sup:3pja7b27])/dt = 4? ? 2r ? dr/dt

dV/dt = c(dA/dt) ? (4/3)? ? 3r[sup:3pja7b27]2[/sup:3pja7b27] ? dr/dt = c[4? ? 2r ? dr/dt] ? r = 2c

I'm not sure how this result connects with dr/dt being a constant. Am I on the right track, and where do I go from here?
 
You're done. c is a constant. If r is a multiple of c, r is constant.

Good work.

It looks like you may not have noticed that dV/dt = Surface Area. I wonder if that's a coincidence?
 
Thanks! The problem statement wanted me to show that dr/dt is a constant, not just that r is a constant. So if r is a constant, then dr/dt is necessarily a constant? I suppose so, since dr/dt = 0. Is that how I should complete my answer?
 
As a spherical mothball evaporates, its volume decreases at a rate proportional to its surface area. Show that the rate of decrease of the radius is constant.

Here's what I've done so far:

Let r be the radius of the sphere, V its volume, and A its surface area. We know V = (4/3)?r^3 and A = 4?r^2. We are given dV/dt = c(dA/dt), where c is a nonzero constant. We want to show that dr/dt is a constant.
Click to expand...

I believe you have a misinterpretation of the problem statement: "volume decreases at a rate proportional to its surface area." That means dV/dt = cA, NOT dV/dt = c(dA/dt). "r" is not a constant if the ball is evaporating. A constant r would mean no evaporation.
 
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