1. **Problem statement:** We need to find the height of point A above the ground on a roller coaster drop of 65 m horizontal distance and 65 m vertical height, divided into two sections with gradients 1/2 and 3. 2. **Understanding gradients:** Gradient (slope) is defined as $\text{gradient} = \frac{\text{vertical change}}{\text{horizontal change}}$. 3. **Given:** Total horizontal distance = 65 m, total vertical height = 65 m. 4. **Let:** The horizontal distance from the start to point A be $x$ meters. 5. **First section gradient:** $\frac{\text{height at A}}{x} = \frac{1}{2}$, so height at A is $\frac{x}{2}$. 6. **Second section gradient:** The remaining horizontal distance is $65 - x$ meters. 7. The vertical height from A to the top is $65 - \frac{x}{2}$ (since total height is 65 and height at A is $\frac{x}{2}$). 8. Using the gradient of the second section: $$3 = \frac{65 - \frac{x}{2}}{65 - x}$$ 9. Multiply both sides by $65 - x$: $$3(65 - x) = 65 - \frac{x}{2}$$ 10. Expand left side: $$195 - 3x = 65 - \frac{x}{2}$$ 11. Bring all terms to one side: $$195 - 3x - 65 + \frac{x}{2} = 0$$ 12. Simplify constants: $$130 - 3x + \frac{x}{2} = 0$$ 13. Combine $-3x + \frac{x}{2} = -\frac{6x}{2} + \frac{x}{2} = -\frac{5x}{2}$: $$130 - \frac{5x}{2} = 0$$ 14. Solve for $x$: $$\frac{5x}{2} = 130$$ $$5x = 260$$ $$x = 52$$ 15. Find height at A: $$\text{height at A} = \frac{x}{2} = \frac{52}{2} = 26$$ **Final answer:** The height of point A above the ground is 26 meters.