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## Homework Statement

A pond forms as water collects in a conical depression of radius

*a*and depth

*h*. Suppose water flows in at a constant rate,

*k*and is lost through evaporation at a rate proportional to the surface area.

I was wondering whether anyone could give me some guidance on this part of the question:

Find the equilibrium depth of water in the pool. Is the equilibrium asymptotically stable?

## Homework Equations

[tex]\frac{dV}{dt}=k-\alpha \pi(\frac{3aV}{\pi h})^{\frac{2}{3}}[/tex]

Where [tex]\alpha[/tex] = coefficient of evaporation.

radius at any time r(t) and depth at any time l(t).

[tex]\frac{a}{h}=\frac{r}{l}[/tex]

## The Attempt at a Solution

[tex]\frac{dV}{dt}=0=k-\alpha \pi(\frac{3aV}{\pi h})^{\frac{2}{3}}[/tex]

Putting the differential equation only in terms of radius and alpha:

[tex]\alpha \pi r^2 =k[/tex]

[tex]r=\sqrt{(\frac{k}{\alpha \pi})}[/tex]

given that [tex] l=r\cdot\frac{h}{a}[/tex],

[tex]l=\frac{h}{a}\sqrt{(\frac{k}{\alpha \pi})}[/tex]

But I am not sure how to find whether or not this is asymptotically stable.