x varies directly as y squared, x = 12 when y = 2, find x when y = 5.

Use the ratio method: $(x_1/x_2) = (y_1/y_2)^2$. Thus, $12/x_2 = (2/5)^2 = 4/25$. Therefore, $x_2 = 12 \times (25/4) = 75$.

Students often forget to square the variable $y$ and simply set up a direct proportion like $12/2 = x/5$, which leads to the incorrect answer 30.

This concept of direct variation is foundational for Physics problems involving relationships like $E \propto v^2$ (Kinetic Energy) or $F \propto r^{-2}$ (Inverse Square Law), where scaling factors allow for rapid mental calculation.