Two curved lanes in a park design are modelled by y = x² and y = x³, with lengths in metres…

Step 1. The points. $x^2 = x^3$ gives $x^2(1 - x) = 0$, so $x = 0$ or $x = 1$. The points are $(0, 0)$ and $(1, 1)$.

Step 2. At $(1, 1)$. The gradients are $m_1 = 2$ and $m_2 = 3$. $\tan\varphi = \left|\dfrac{2 - 3}{1 + 6}\right| = \dfrac{1}{7}$, so $\varphi = 8.13^\circ$.

Step 3. At the origin. Both gradients are $0$, so $\tan\varphi = 0$ and $\varphi = 0^\circ$.

What it means. At the origin the two lanes have the same tangent, the $x$-axis. They touch and run side by side there. At $(1, 1)$ they cross at a small angle.

The rule. $8.13^\circ$ is less than $30^\circ$, so the lanes do not meet the rule at $(1, 1)$.