Christmas Math Problem 2

Geometry Level pending

In Santa's workshop, there is a large, rectangular room. All of the points on this rectangle are labelled \(A\), \(B\), \(C\), and \(D\). \(A\) and \(B\) are the two points on the corners on the floor, and \(C\) and \(D\) are the ones on the roof. We are going to assume that this room is two-dimensional, so there are only four points. Inside of this rectangular room, there is a skating rink across the floor, which is used by all of the elves in Santa's workshop. This skating rink stretches all the way from point \(A\) to point \(B\), across the floor. Above this skating rink is a slide, and a ramp leading up to the slide. This slide and ramp stretch all the way over the skating rink, ending at the points \(A\) and \(B\). We will call the point where the ramp ends and the slide starts going down as \(Y\). We can measure that the side length of the slide, or line \(AY\) is equal to 15, the length of the ramp or line \(BY\) is 18 and the angle between the \(AY\) and \(BY\) at point \(Y\) is equal to \(57\) degrees. If an elf skating across the rink starts at point \(A\), and their acceleration is modelled by the differential equation, \(a(x)\) = \(3x^2 + 6\), where \(x\) is the elf's position, and \(a(x)\) is the acceleration, and we know that \(v(x)\), which is velocity, at \(v(2)\) is equal to \(9\), then what is the elf's velocity when it is at point \(B\), rounded to the nearest hundredth?

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