Question

The base of an auditorium is in the form of an eclipse 200 feet long and 100 feet wide a pin drop near one focus can clearly be heard at the other focus determine the distance between the foci to the nearest 10th of a foot.... how would I work this out??

Answer 1

The distance between the foci of an elliptical auditorium, with major and minor axes of 200 feet and 100 feet respectively, can be calculated using the formula c = √(a² - b²). The calculation shows that the distance between the two foci is approximately 173.2 feet.

Step-by-step explanation:

To determine the distance between the foci of an elliptical auditorium, you first need to understand the properties of an ellipse. The major axis is the longest diameter of an ellipse, and in this case, it is 200 feet long. The minor axis is the shortest diameter, and it is given as 100 feet wide. The distance between the foci of an ellipse, denoted as 2c, can be found using the equation c = √(a² - b²), where a is half the length of the major axis, and b is half the length of the minor axis.

In this particular question, a = 200 feet / 2 = 100 feet and b = 100 feet / 2 = 50 feet. Plugging these values into the equation gives us:

c = √(100² - 50²)

c = √(10000 - 2500)

c = √7500

c ≈ 86.6 feet

Therefore, the distance between the two foci (2c) is approximately 173.2 feet to the nearest tenth of a foot.

Answer 2

Let the coordinate of focus be
`(\pm c , 0)`

As per the statement: The base of an auditorium is in the form of an eclipse 200 feet long and 100 feet wide.

⇒Length of Major axis=base of an auditorium = 200 feet and Length of a minor axis=wide of a auditorium = 100 ft

Semi-major axis (a) = 100 ft and

semi-minor axis(b) = 50 ft

Then, by an equation:

`c^2 = a^2-b^2`

Solve for c:

Substitute the given values we have;

`c^2=(100)^2-(50)^2`

Simplify:

`c^2 = 7500`

or

`c=√(7500) = 86.6025404` ft

Distance between the foci is,
`2c = 2 \cdot 86.6025404 = 173.205081`

Therefore, the distance between the foci to the nearest 10th of a foot is, 173.2 ft