Chapter 14, Problem 48RE

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Surface Area The roof over the stage of an open air theater at a theme park is modeled by f ( x , y ) = 25 [ 1 + e − ( x 2 + y 2 ) / 1000 cos 2 ( x 2 + y 2 1000 ) ] where the stage is a semicircle bounded by the graphs of y = 50 2 − x 2 and y = 0 .(a) Use a computer algebra system to graph the surface.(b) Use a computer algebra system to approximate the number of square feet of roofing required to cover the surface.

(a)

To determine

To graph: The roof over the stage of an open air theater modeled by f(x,y)=25[1+e(x2+y2)1000cos2((x2+y2)1000)], where the stage is bounded by graphs of y=502x2 and y=0 and is semicircular.

Explanation

Given:

Roof of an open air theater is modeled by,

f(x,y)=25[1+eâˆ’(x2+y2)1000cos2((x2+y2)1000)], where the stage is bounded by graphs of y=502âˆ’x2 and y=0 and is semicircular.

Formula used:

Conversion of Cartesian coordinate to cylindrical coordinate,

r2=x2+y2

Graph:

Take the provided model of the roof of an open air theater into consideration.

f(x,y)=25[1+eâˆ’(x2+y2)1000â‹…cos2((x2+y2)1000)]

Now, on substituting the formula r2=<

(b)

To determine

To calculate: The square foot of roofing required to cover the surface plotted in part (a) with the help of computer algebra system.

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