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Q 71.

Expert-verifiedFound in: Page 655

Book edition
6th

Author(s)
Sullivan

Pages
1200 pages

ISBN
9780321795465

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 $43.3feet$.

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 $100feet$.

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

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.

${b}^{2}={50}^{2}-{25}^{2}\phantom{\rule{0ex}{0ex}}{b}^{2}=1875\phantom{\rule{0ex}{0ex}}b=\sqrt{1875}\phantom{\rule{0ex}{0ex}}b=43.3$

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

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