1. **State the problem:** We are given a cubic function $h(x) = x^3 - 3x$ representing a water slide profile, and two points $A(2,3)$ and $B(6,9)$ defining a line segment. We want to analyze the line segment $AB$ and find the perpendicular line from the origin $(0,0)$ to this line. 2. **Find the equation of the line passing through points $A$ and $B$:** The slope $m$ of the line through $A(2,3)$ and $B(6,9)$ is $$m = \frac{9 - 3}{6 - 2} = \frac{6}{4} = \frac{3}{2}.$$ Using point-slope form with point $A$: $$y - 3 = \frac{3}{2}(x - 2).$$ Simplify: $$y = \frac{3}{2}x - 3 + 3 = \frac{3}{2}x.$$ So the line $AB$ has equation: $$y = \frac{3}{2}x.$$ 3. **Find the equation of the perpendicular line from the origin to line $AB$:** The slope of line $AB$ is $\frac{3}{2}$, so the slope of the perpendicular line is the negative reciprocal: $$m_\perp = -\frac{2}{3}.$$ Since it passes through the origin $(0,0)$, its equation is: $$y = -\frac{2}{3}x.$$ 4. **Find the point of intersection $P$ between the line $AB$ and the perpendicular line:** Set the two equations equal: $$\frac{3}{2}x = -\frac{2}{3}x.$$ Bring all terms to one side: $$\frac{3}{2}x + \frac{2}{3}x = 0.$$ Find common denominator 6: $$\frac{9}{6}x + \frac{4}{6}x = \frac{13}{6}x = 0.$$ This implies $x=0$, but $x=0$ corresponds to the origin, which is on the perpendicular line but not on $AB$ (since $y=\frac{3}{2}x$ gives $y=0$ at $x=0$). This suggests the origin lies on the line $AB$ only if $y=0$, but $AB$ passes through $(2,3)$ and $(6,9)$, so origin is not on $AB$. Re-examining the intersection: Set $$\frac{3}{2}x = -\frac{2}{3}x$$ $$\Rightarrow \frac{3}{2}x + \frac{2}{3}x = 0$$ $$\Rightarrow \left(\frac{9}{6} + \frac{4}{6}\right)x = 0$$ $$\Rightarrow \frac{13}{6}x = 0$$ $$\Rightarrow x=0.$$ At $x=0$, $y=0$ for both lines, so the origin is the intersection point. 5. **Check if the origin lies on line $AB$:** Plug $x=0$ into $y=\frac{3}{2}x$: $$y=0.$$ So the origin $(0,0)$ lies on line $AB$. 6. **Conclusion:** The perpendicular line from the origin to line $AB$ is the origin itself, meaning the origin lies on the line $AB$. Therefore, the perpendicular distance from the origin to line $AB$ is zero. **Final answer:** - Equation of line $AB$: $$y = \frac{3}{2}x$$ - Equation of perpendicular line from origin: $$y = -\frac{2}{3}x$$ - Intersection point is the origin $(0,0)$, so the origin lies on line $AB$. - Perpendicular distance from origin to line $AB$ is zero.