Question

An elliptical-shaped whispering gallery has two chairs positioned at the foci of the ellipse, 6 meters apart. The width of the room, measured along it's minor axis, is 8 meters. Find an equation of the ellipse representing the floor of the room if its center is at the origin and its major axis is horizontal. What is the length of the room measured along its major axis?equation: _____________ length of room: ______________

Answer 1

Given:

distance between Foci = 6 meters

Minor axis = 8 meters

Since distance is 6, f = 3

Minor axis is 8, b = 4

Let's use the Foci formula:

`f\text{ = }\sqrt[]{a^2-b^2}`

Solve for a:

`\begin{gathered} 3\text{ = }\sqrt[]{a^2-4^2} \\ \\ 3^2=(\sqrt[]{a^2-4^2})^2 \\ \\ 9=a^2-16 \\ \\ 9+16=a^2 \\ \\ 25=a^2 \\ \\ \sqrt[]{25}=a \\ \\ a\text{ = 5} \end{gathered}`

The equation will be:

`(x^2)/(5^2)+(y^2)/(4^2)=1`

The length of the room is.

We have the vertices as:

Vertex, 1: (5, 0)

Vertex 2: (-5, 0)

The Length of the room = 5 + 5 = 10 meters

ANSWER:

Equation:

`(x^2)/(5^2)+(y^2)/(4^2)=1`

Length = 10 meters