The rotunda of the Capitol building in Washington, D.C is a “whispering room” -

Question

The rotunda of the Capitol building in Washington, D.C is a “whispering room” - one with an elliptically-arched ceiling. If Maria stands at one focus of the ellipse and whispers to her friend, the dispersed sound waves are reflected by the ceiling and concentrated at the other focus, allowing people across the room to clearly hear what Maria said. Suppose such a gallery has a ceiling reaching twenty feet above the five-foot-high vertical walls at its tallest point (so the cross-section is half an ellipse topping two 5-foot high vertical lines at either end), and suppose the foci of the ellipse are thirty feet apart. What is the height of the ceiling above each “whispering point” (foci)?

Explanation / Answer

center of the ellipse = (0,5)

and foci of the ellipse are 30 feet apart

so, c = 30/2 = 15

we have vertical wals at its tallest point is 20 feet

b = 20

now finding a

a^2 = b^2 +c^2

a^2 = 20^2 + 15^2

a = 25

so equation of the ellipse is

x^2 / 25^2 + ( y - 5)^2 / 20^2 = 1

height of the ceiling above each whispering point

that is plug x = 15

15^2 / 25^2 + ( y-5)^2 / 20^2 = 1

225 / 625 + ( y - 5)^2 / 400 = 1

144 + ( y -5 )^2 = 400

(y-5)^2 = 256

y - 5 = 16

y = 21

hence, height of the ceiling is 21 feet

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.