Solve V=1/9pik^2(9N-k)

$$V = \frac { 1 } { 9 } \pi k ^ { 2 } ( 9 N - k )$$

Solve for N

$$\left\{\begin{matrix}N=\frac{k}{9}+\frac{V}{\pi k^{2}}\text{, }&k\neq 0\\N\in \mathrm{R}\text{, }&V=0\text{ and }k=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{9}\pi k^{2} by 9N-k.

$$V=\pi k^{2}N-\frac{1}{9}\pi k^{3}$$

Swap sides so that all variable terms are on the left hand side.

$$\pi k^{2}N-\frac{1}{9}\pi k^{3}=V$$

Add \frac{1}{9}\pi k^{3} to both sides.

$$\pi k^{2}N=V+\frac{1}{9}\pi k^{3}$$

The equation is in standard form.

$$\pi k^{2}N=\frac{\pi k^{3}}{9}+V$$

Divide both sides by \pi k^{2}.

$$\frac{\pi k^{2}N}{\pi k^{2}}=\frac{\frac{\pi k^{3}}{9}+V}{\pi k^{2}}$$

Dividing by \pi k^{2} undoes the multiplication by \pi k^{2}.

$$N=\frac{\frac{\pi k^{3}}{9}+V}{\pi k^{2}}$$

Divide V+\frac{\pi k^{3}}{9} by \pi k^{2}.

$$N=\frac{k}{9}+\frac{V}{\pi k^{2}}$$

Solve for V

$$V=\frac{\pi \left(9N-k\right)k^{2}}{9}$$