(Originais Teachy 2023) - Question Very Hard of Mathematics

An engineer is designing the ramp of a parking lot, which must have the shape of a parabolic arch to ensure that cars go up smoothly and continuously. The equation that models the parabola is given by y = -x^2 + 6x, where x represents the horizontal distance from the origin point and y the height in meters. The engineer needs to determine the maximum height that the ramp reaches, as this will affect the design of the parking lot's roof. To do so, he must find the vertex of the parabola, which corresponds to the point of maximum height. Considering the second-degree equation y = -x^2 + 6x, identify the value of x that determines the maximum height of the parabola and then calculate the maximum height that the ramp reaches. Discuss the practical implication of this value for the ramp's design and elaborate on how solving this equation can be applied in real engineering and architecture projects to ensure the safety and comfort of users.