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## Homework Statement

Solve the equation for [itex]r,r>0,r<R[/itex].

[tex]\frac{-2\pi R^3}{3}-\frac{8\pi r^2\sqrt{R^2-r^2}}{3}+2r^2R+\frac{2\pi}{3}R^2\sqrt{R^2-r^2}=\frac{2\pi R^3}{3}[/tex]

**2. The attempt at a solution**

After factoring,

[tex][/tex]

[tex]-2R(2\pi R^2-3r^2)=2\pi\sqrt{R^2-r^2}(4r^2-R^2).[/tex]

After squaring,

[tex]64\pi^2 r^6+(36R^4-96\pi^2 R^2)r^4+(32\pi^2 R^4-44\pi^2 R^4)r^2+12\pi^2 R^6=0.[/tex]

[itex]r[/itex] is unknown variable, so the last equation should be the equation of six degree polynomial.

How to solve it?