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Q 71.

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Precalculus Enhanced with Graphing Utilities

Found in: Page 655

Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

Whispering Gallery: A hall $100$ feet in length is to be designed as a whispering gallery. If the foci are located $25$ feet from the center, how high will the ceiling be at the center?

The height of the ceiling at the center of the gallery is $43.3 f e e t$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The center of the ellipse be the origin and the major axis be along the x-axis.

Here, the center of the room is at $(0, 0)$ , It is given that the length of the room is $100 f e e t$ .

Step 2. Height of the ceiling at the center.

The distance from the center of the room to each vertex is $\frac{100}{2} = 50 f e e t$

The distance from center to each focus is $c = 25$

Substitute $a = 50$ and $c = 25$ in $b^{2} = a^{2} - c^{2}$ we get.

$b^{2} = 50^{2} - 25^{2} b^{2} = 1875 b = \sqrt{1875} b = 43.3$

Thus, the height of the ceiling at the center of the gallery is $43.3$ feet.

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