A hall 200 feet in length is to be designed as an ellipse if the foci are 35 feet from the center. How high will the ceiling be from the center?
To find the height of the ceiling from the center of the ellipse, we need to determine the semi-major axis (a) and the semi-minor axis (b) of the ellipse.
In an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant. This property is known as the definition of an ellipse.
Given that the foci are 35 feet from the center, the sum of the distances from any point on the ellipse to the two foci is 2 * 35 = 70 feet.
We are also given the length of the hall, which is the total length of the major axis (2a). So, 2a = 200 feet.
To find the value of 'a', we can divide the total length of the major axis by 2: a = 200 / 2 = 100 feet.
Now, let's find the semi-minor axis 'b'. The distance from the center of the ellipse to each focus is 'c' (given as 35 feet).
Using the formula for the relationship between 'a', 'b', and 'c' in an ellipse,
c^2 = a^2 - b^2
Substituting the values, we have:
35^2 = 100^2 - b^2
1225 = 10000 - b^2
b^2 = 10000 - 1225
b^2 = 8775
b = √8775 ≈ 93.7 feet
Finally, to find the height of the ceiling from the center, we can subtract the semi-minor axis (b) from the semi-major axis (a):
Ceiling height from the center = a - b = 100 - 93.7 ≈ 6.3 feet
Therefore, the height of the ceiling from the center will be approximately 6.3 feet.