The ceiling in a hallway 20 ft wide is in the shape of a semi ellipse and 18 ft high - Maths - Conic Sections

Question

The ceiling in a hallway 20 ft wide is in the shape of a semi
ellipse and 18 ft high at the centre Find the height of the ceiling
4 ft from either wall if the height of the side wall is 12ft

Naveen Bharadwaj · Answer

Let PQR be the height of the ceiling which is 4 feet from the
wall From the diagram PQ = 12 ft To find the height QR
Since the width is 20ft, take A, A′ as vertices with A as
(10, 0) and A′ as ( 10, 0) Take the midpoint of AA′ as
the centre which is (0, 0)
AA′ = 2a = 20 ⇒ a = 10
and b = 18 12 = 6
∴x2/100 +y2/36 = 1
Let QR be y1 then R is (6, y1) Since R lies on the ellipse,
36/100 +y12/36 = 1 ⇒ y1 = 4 8
∴ PQ + QR = 12 + 4 8
∴ The required height of the ceiling is 16 8 feet

The ceiling in a hallway 20 ft wide is in the shape of a semi ellipse and 18 ft high at the centre.Find the height of the ceiling 4 ft from either wall if the height of the side wall is 12ft.