Chapter 14, Problem 45RE

Building Design A new auditorium is built with a foundation in the shape of one-fourth of a circle of radius50 feet. So, it forms a region R bounded by the graph of $x^2+y^2={50}^2\text{with}x\ge 0\text{and }x\ge 0$ The following equations are models for the floor and ceiling.

$\begin{array}{l}\text{Floor:}z=\frac{x+y}{5}\\ \text{ Ceiling:}\text{z=20+}\frac{xy}{100}\end{array}$

(a) Calculate the volume of the room, which is needed to determine the heating and cooling requirements.

(b) Find the surface area of the ceiling.

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