An architect wishes to design a large room 122 feet long that will be a whispering gallery. The ceiling of the room has a cross section that is an ellipse whose foci are to be located 14 feet to the right and the left of center. Find the height h of the elliptical ceiling ( to the nearest tenth of a foot.)
in an ellipse with semi axes a and b, the foci are at distance c from the center, where
a^2 = b^2 + c^2
Here, a = 61 and c = 14
The height will be the semi-minor axis, b
61^2 = b^2 + 14^2
b^2 = 3525
b = 59.4
To find the height (h) of the elliptical ceiling, we need to use the equation of an ellipse:
(x^2 / a^2) + (y^2 / b^2) = 1,
where a is the semi-major axis and b is the semi-minor axis.
In this case, we know that the foci (f1 and f2) of the ellipse are located 14 feet to the right and left of the center. Therefore, we can use the relationship:
2a = distance between the foci (f1f2).
Since the distance between the foci is 28 feet (14 feet to the right + 14 feet to the left), we have:
2a = 28,
a = 28 / 2,
a = 14.
Now, we need to find the value of b.
As the room is 122 feet long, the length of the major axis is 122 feet, so 2a is equal to this length:
2a = 122,
a = 122 / 2,
a = 61.
To find b, we can use the formula:
b = sqrt(a^2 - c^2),
where c is the distance between the center and one of the foci (c = 14 feet).
b = sqrt(61^2 - 14^2),
b = sqrt(3721 - 196),
b = sqrt(3525).
Now, we have the values of a and b.
Finally, we can find the height (h) of the elliptical ceiling, which is equal to 2b:
h = 2b,
h = 2(sqrt(3525)),
h ≈ 118.5 feet.
Therefore, the height of the elliptical ceiling is approximately 118.5 feet.