Solve c=t^2/200(200-t^2)

$$c = \frac { t ^ { 2 } } { 200 } ( 200 - t ^ { 2 } )$$

Solve for c

$$c=-\frac{t^{4}}{200}+t^{2}$$

Solution Steps

Express \frac{t^{2}}{200}\left(200-t^{2}\right) as a single fraction.

$$c=\frac{t^{2}\left(200-t^{2}\right)}{200}$$

Use the distributive property to multiply t^{2} by 200-t^{2}.

$$c=\frac{200t^{2}-t^{4}}{200}$$

Divide each term of 200t^{2}-t^{4} by 200 to get t^{2}-\frac{1}{200}t^{4}.

$$c=t^{2}-\frac{1}{200}t^{4}$$

Solve for t (complex solution)

$$t=\sqrt{10\sqrt{100-2c}+100}$$

$$t=-\sqrt{10\sqrt{100-2c}+100}$$

$$t=-\sqrt{-10\sqrt{100-2c}+100}$$

$$t=\sqrt{-10\sqrt{100-2c}+100}$$

Solve for t

$$\left\{\begin{matrix}t=-\sqrt{-10\sqrt{100-2c}+100}\text{; }t=\sqrt{-10\sqrt{100-2c}+100}\text{, }&c\geq 0\text{ and }c\leq 50\\t=-\sqrt{10\sqrt{100-2c}+100}\text{; }t=\sqrt{10\sqrt{100-2c}+100}\text{, }&c\leq 50\end{matrix}\right.$$