with detailed solution

You are an engineer who works for Calculus City, a theme park for Mathematicians. You are assigned by the Head Engineer to innovate the ride, Derivative F lyer. by designing its initial nerve-racking ascent and drop. After studying the structure of Derivative F iyer, you want that the gradient of the ascent to be % and the gradient of the drop to be ?8' You decide to connect these two straight paths represented by y = Pl(x)and y = P2 (x) with part of a parabola with an equation, f (x) = pxz + qx + r . All measurements are in feet. You also want the track of Derivative F Iyer to be smooth by assuring that there are no sudden changes in the direction of the ride, this can only be achieved if the paths P, (x) and P2 (x) are tangent to parabola at the transition points A and B. By default, you plot that the point A be at the origin. 1. Suppose that the horizontal distance from A and B is 100 ft. Write equations in p, q and / that will guarantee that the track of Derivative Flyer is indeed smooth at the transition points. 1i. Solve the equations in (i) for p, q and r to find a formula for f(x). Use Desmos to also plot the graph of Pi(x), f and P2(x) iii. Find the difference in the elevation between A and B