# 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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