X,Y and Z are related quantities such that x varies directly as y and inversely as the square root of z. When x=300 and y=65, z=25. Calculate the value of x when Y= 468 and Z =144.

"x varies directly as y and inversely as the square root of z" means that $x$ is proportional to $y$ and inversely proportional to $\sqrt{z}$. We introduce a constant of proportionality, $k$.

$$x = \frac{ky}{\sqrt{z}}$$

We are given $x=300$ when $y=65$ and $z=25$. Substitute these values into the equation:

$$300 = \frac{k(65)}{\sqrt{25}}$$ $$300 = \frac{65k}{5}$$ $$300 = 13k$$

Divide by 13 to solve for $k$:

$$k = \frac{300}{13}$$

$$x = \frac{\frac{300}{13}y}{\sqrt{z}}$$ $$x = \frac{300y}{13\sqrt{z}}$$

Substitute $y=468$ and $z=144$ into the equation:

$$x = \frac{300(468)}{13\sqrt{144}}$$ $$x = \frac{300 \times 468}{13 \times 12}$$ $$x = \frac{140400}{156}$$ $$x = 900$$

The value of $x$ when $y=468$ and $z=144$ is $\boxed{\textbf{900}}$.