A new auditorium is built with a foundation in the shape

Chapter 14, Problem 45

(choose chapter or problem)

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

\(x^{2}+y^{2}=50^{2}\)

with \(x \geq 0\) and \(y \geq 0\). The following equations are models for the floor and ceiling.

Floor: \(z=\frac{x+y}{5}\)

Ceiling: \(z=20+\frac{x y}{100}\)

(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.

Text Transcription:

x^2 + y^2 = 50^2

x geq 0

y geq 0

z = x + y / 5

z = 20 + xy / 100