#### Answer

$\color{blue}{y=\frac{1}{2}x^2}$

#### Work Step by Step

RECALL:
(1) When $y$ varies directly as $x$, the variation is direct and is represented by the equation $y=kx$ where $k$ is the constant of variation.
(2) When $y$ varies inversely as $x$, the variation is inverse and is represented by the equation $y=\frac{k}{x}$ where $k$ is the constant of variation.
$y$ varies directly as he square of $x$ so the equation is $y=kx^2$ with $k$=constant of variation. To find the value of $k$, substitute the given values into $y=kx^2$ to obtain:
$$y=kx^2
\\50=k(10^2)
\\50 =k(100)
\\\frac{50}{100}=\frac{k(100)}{100}
\\\frac{1}{2}=k$$
Thus, the equation of the variation is: $\color{blue}{y=\frac{1}{2}x^2}$.