Answer to Question #124582 in Analytic Geometry for Samuel kassapa

Question

A statuary hall is elliptical in shape. It measures 42 feet wide and 96 feet long. If a person is standing at one focus, her whisper can be heard by a person standing at the other focus. How far apart are the two people

Answer 1

As per the question the elliptical shaped statuary hall has been drawn which is 96 ft long and 48 ft

wide. Let us assume that the person is standing at focus F₁ and her whisper can be heard by a person standing at another focus F₂.

This a typical ellipse with major axis 96 ft and Minor axis 42 ft following standard equation of ellipse which is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$

Here $a= 48 \space and \space b=21\\\therefore eccenticity (e) = \sqrt{1-\frac{b^2}{a^2}}$

$Now \space e = \sqrt{1-\frac{21^2}{48^2}}\\or, e=\sqrt{1-\frac{441}{2304}}\\or, e=\sqrt{\frac{2304-441}{2304}}\\or, e=\sqrt{\frac{1863}{2304}}\\or, e=\frac{\sqrt{1863}}{48}\\or, e=\frac{43.16}{48}\\or, e = 0.899 \\or, e = 0.9$

now the distance between the foci F₁ and F₂ is F₁F₂ $=2ae.$

$\therefore 2ae = 2*48*0.9$

$=$ $86.4 \space ft.$

$\therefore$ The two persons are 86.4 ft apart. (Answer)