Answer
$x = \frac{2}{27}$
Work Step by Step
Slope of line $x+2y-2 = 0$ is $m_{1} = -\frac{1}{2}$
So, the line perpendicular has slope $m = 2$ (negative reciprocal)
Slope of curve $y^{3} = 2x^{2}$ can be obtained through implicit differentiation :
$3y^{2}\frac{dy}{dx} = 4x$
$\frac{dy}{dx} = \frac{4x}{3y^{2}}$
Set $\frac{dy}{dx} = 2$
$\frac{4x}{3y^{2}} = 2$
$x = \frac{3}{2}y^{2}$
Use the value of x above in equation of curve $y^{3} = 2x^{2}$
$y^{3} = 2((\frac{3}{2})y^{2})^{2}$
$y^{3} = 2(\frac{9}{4}y^{4})$
$y^{3} = \frac{9}{2}y^{4}$
$\frac{2}{9} = y$
or $y = \frac{2}{9}$
Now
as $x =\frac{3}{2}y^{2}$
Putting value of y
$x = (\frac{3}{2})(\frac{2}{9})^{2}$
$x = (\frac{3}{2})(\frac{4}{81})$
$x = \frac{2}{27}$
