Question

A tunnel through a mountain for a four-lane highway is to have a elliptical opening. The total width of the highway (not the opening) is to be 16 m, and the height at the edge of the road must be sufficient for a truck 4 m high to clear if the highest point of the opening is to be 5 m approximately. How wide must the opening be?

Answer 1

Let the equation of the ellipse be

`x^2/"a"^2 + y^2/"b"^2` = 1

Length of semi minor axis b = 5

i.e., `x^2/"a"^2 + y^2/5^2` = 1

Let BB’ be the road width and AA’ be the endpoints of the opening of the tunnel.

Let CB = 8, BD = 4

∴ D is (8, 4) lies on the ellipse

`8^2/"a"^2 + 4^2/5^2` = 1

⇒ a² = `25/9 xx 64`

⇒ a = `40/3`

The width AA’ = 2a

= `80/3`

= 26.66 m

The required width is 26.66 m