A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius 50 feet. So, it forms a region R bounded by the graph of x2 + y2 = 502 with x ≥ 0 and y ≥ 0. The following equations are models for the floor and ceiling. Floor: z = x + y /5 Ceiling: z = 20 + xy /100 Calculate the volume of the room, which is needed to determine the heating and cooling requirements.

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