3. Whispering Gallery. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. This room, sometimes called the Whispering Gallery is \( 46 \mathrm{ft} \). wide and \( 96 \mathrm{ft} \). long. John Quincy Adams is said to have used the focusing properties of the room to overhear conversations.
a. Find an equation that models the shape of the room.
b. Where should the people stand to hear each other (that is - find the foci)

Answer

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Answer

\((-\sqrt{2025}, 0)\), (\sqrt{2025}, 0)

Steps

Step 1 ：\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), a = 48, b = 23

Step 2 ：\(\frac{x^2}{2304} + \frac{y^2}{529} = 1\)

Step 3 ：Foci: \((-\sqrt{a^2-b^2}, 0)\), \((\sqrt{a^2-b^2}, 0)\)

Step 4 ：\((-\sqrt{2025}, 0)\), (\sqrt{2025}, 0)