Whispering Gallery Wonders: Calculating Ceiling Height in Elliptical Marvels
Whispering Gallery: Height of the Ceiling
Explore the intriguing properties of whispering galleries and elliptical shapes. In a whispering gallery, sound waves reflect off the ceiling, allowing communication between foci. Let's find the height of the ceiling from the center of the gallery.
Step 1: Understand the Given Parameters
Length of Whispering Gallery (2a): 110 feet
Distance from Center to Focus (c): 30 feet
Step 2: Calculate Half the Length of the Whispering Gallery (a)
Using the given information:
\[
a = \frac{110}{2} = 55 \text{ feet}
\]
Step 3: Use Ellipse Property to Find the Height of the Ceiling (b)
Using the relationship \(c^2 = a^2 - b^2\), substitute the values:
\[
30^2 = 55^2 - b^2
\]
\[
b^2 = 3025 - 900
\]
\[
b \approx \sqrt{2125} \approx 46.0977 \text{ feet}
\]
Conclusion:
The height of the ceiling from the center of the whispering gallery is approximately \(46.0977\) feet.
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