Solve V=frac{operatorname{loh}(h)}{3}+frac{operatorname{loh}(h)}{3}

$$V = \frac { \operatorname { loh } ( h ) } { 3 } + \frac { \operatorname { loh } ( h ) } { 3 }$$

Solve for V

$$V=\frac{2loh^{2}}{3}$$

Solution Steps

Multiply h and h to get h^{2}.

$$V=\frac{loh^{2}}{3}+\frac{lohh}{3}$$

Multiply h and h to get h^{2}.

$$V=\frac{loh^{2}}{3}+\frac{loh^{2}}{3}$$

Combine \frac{loh^{2}}{3} and \frac{loh^{2}}{3} to get \frac{2}{3}loh^{2}.

$$V=\frac{2}{3}loh^{2}$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=-\frac{l^{-\frac{1}{2}}o^{-\frac{1}{2}}\sqrt{6V}}{2}\text{; }h=\frac{l^{-\frac{1}{2}}o^{-\frac{1}{2}}\sqrt{6V}}{2}\text{, }&o\neq 0\text{ and }l\neq 0\\h\in \mathrm{C}\text{, }&\left(l=0\text{ or }o=0\right)\text{ and }V=0\end{matrix}\right.$$

Solve for h

$$\left\{\begin{matrix}h=\frac{\sqrt{\frac{6V}{lo}}}{2}\text{; }h=-\frac{\sqrt{\frac{6V}{lo}}}{2}\text{, }&\left(V\geq 0\text{ and }o<0\text{ and }l<0\right)\text{ or }\left(V\geq 0\text{ and }l>0\text{ and }o>0\right)\text{ or }\left(V\leq 0\text{ and }l<0\text{ and }o>0\right)\text{ or }\left(V\leq 0\text{ and }o<0\text{ and }l>0\right)\\h\in \mathrm{R}\text{, }&\left(l=0\text{ or }o=0\right)\text{ and }V=0\end{matrix}\right.$$