1. The problem gives the height of a roller coaster cart as a function of time: $$y = -0.05(x - 55.5)^2 + 154$$ with domain $$0 < x \leq 111$$. 2. This is a quadratic function in vertex form, where the vertex is at $$x = 55.5$$ and $$y = 154$$. 3. Since the coefficient of the squared term is negative ($$-0.05$$), the parabola opens downward, meaning the vertex represents the maximum height. 4. Therefore, the maximum height the cart reaches is $$154$$ feet, occurring at time $$x = 55.5$$ seconds. 5. The final height of the cart at $$x = 111$$ seconds can be found by substituting $$x = 111$$ into the equation: $$y = -0.05(111 - 55.5)^2 + 154 = -0.05(55.5)^2 + 154 = -0.05 \times 3080.25 + 154 = -154.0125 + 154 = -0.0125$$ 6. Since height cannot be negative in this context, the cart is essentially at ground level (height approximately 0) at $$x = 111$$ seconds. 7. Summary: - Maximum height: $$154$$ feet at $$x = 55.5$$ seconds. - Final height at $$x = 111$$ seconds: approximately $$0$$ feet (ground level).