Solve AtA=1/3pir^2h+4/3pir^3

$$A t A = \frac { 1 } { 3 } \pi r ^ { 2 } h + \frac { 4 } { 3 } \pi r ^ { 3 }$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=\frac{3\times \left(\frac{A}{r}\right)^{2}t}{\pi }-4r\text{, }&r\neq 0\\h\in \mathrm{C}\text{, }&r=0\text{ and }\left(A=0\text{ or }t=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$A^{2}t=\frac{1}{3}\pi r^{2}h+\frac{4}{3}\pi r^{3}$$

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

$$\frac{1}{3}\pi r^{2}h+\frac{4}{3}\pi r^{3}=A^{2}t$$

Subtract \frac{4}{3}\pi r^{3} from both sides.

$$\frac{1}{3}\pi r^{2}h=A^{2}t-\frac{4}{3}\pi r^{3}$$

The equation is in standard form.

$$\frac{\pi r^{2}}{3}h=tA^{2}-\frac{4\pi r^{3}}{3}$$

Divide both sides by \frac{1}{3}\pi r^{2}.

$$\frac{3\times \frac{\pi r^{2}}{3}h}{\pi r^{2}}=\frac{3\left(tA^{2}-\frac{4\pi r^{3}}{3}\right)}{\pi r^{2}}$$

Dividing by \frac{1}{3}\pi r^{2} undoes the multiplication by \frac{1}{3}\pi r^{2}.

$$h=\frac{3\left(tA^{2}-\frac{4\pi r^{3}}{3}\right)}{\pi r^{2}}$$

Divide tA^{2}-\frac{4\pi r^{3}}{3} by \frac{1}{3}\pi r^{2}.

$$h=\frac{3tA^{2}}{\pi r^{2}}-4r$$

Solve for h

$$\left\{\begin{matrix}h=\frac{3\times \left(\frac{A}{r}\right)^{2}t}{\pi }-4r\text{, }&r\neq 0\\h\in \mathrm{R}\text{, }&\left(A=0\text{ or }t=0\right)\text{ and }r=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$A^{2}t=\frac{1}{3}\pi r^{2}h+\frac{4}{3}\pi r^{3}$$

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

$$\frac{1}{3}\pi r^{2}h+\frac{4}{3}\pi r^{3}=A^{2}t$$

Subtract \frac{4}{3}\pi r^{3} from both sides.

$$\frac{1}{3}\pi r^{2}h=A^{2}t-\frac{4}{3}\pi r^{3}$$

The equation is in standard form.

$$\frac{\pi r^{2}}{3}h=tA^{2}-\frac{4\pi r^{3}}{3}$$

Divide both sides by \frac{1}{3}\pi r^{2}.

$$\frac{3\times \frac{\pi r^{2}}{3}h}{\pi r^{2}}=\frac{3\left(tA^{2}-\frac{4\pi r^{3}}{3}\right)}{\pi r^{2}}$$

Dividing by \frac{1}{3}\pi r^{2} undoes the multiplication by \frac{1}{3}\pi r^{2}.

$$h=\frac{3\left(tA^{2}-\frac{4\pi r^{3}}{3}\right)}{\pi r^{2}}$$

Divide tA^{2}-\frac{4r^{3}\pi }{3} by \frac{1}{3}\pi r^{2}.

$$h=\frac{3tA^{2}}{\pi r^{2}}-4r$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=-\frac{it^{-\frac{1}{2}}\sqrt{-3\pi \left(4r+h\right)}r}{3}\text{; }A=\frac{it^{-\frac{1}{2}}\sqrt{-3\pi \left(4r+h\right)}r}{3}\text{, }&t\neq 0\\A\in \mathrm{C}\text{, }&\left(r=0\text{ or }r=-\frac{h}{4}\right)\text{ and }t=0\end{matrix}\right.$$

Solve for A

$$\left\{\begin{matrix}A=\frac{\sqrt{\frac{3\pi \left(4r+h\right)r^{2}}{t}}}{3}\text{; }A=-\frac{\sqrt{\frac{3\pi \left(4r+h\right)r^{2}}{t}}}{3}\text{, }&\left(t>0\text{ or }h\leq -4r\right)\text{ and }\left(t<0\text{ or }h\geq -4r\right)\text{ and }t\neq 0\\A\in \mathrm{R}\text{, }&\left(r=0\text{ or }r=-\frac{h}{4}\right)\text{ and }t=0\end{matrix}\right.$$