Question

An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Answer 1

Let O be the vertex of arch. Its equation will be `x^(2)=4ay`.
Given that OC = 10 and `AC=(5)/(2)`
`:.` Coordinates of A are `((5)/(2),10)` which lies on the parabola `x^(2)=4ay`

`((5)/(2))^(2)=4a*(10)rArr4a=(5)/(8)`
`:.x^(2)=(5)/(8)y` . . . (1)
Now, OE = 2
Let EF = h
Coordinates of F=(h,2)
Put in equation (1)
`h^(2)=(5)/(8)xx2=(5)/(4)`
`rArr" "h=(sqrt(5))/(2)`
`:.DF=2EF=2h=2xx(sqrt(5))/(2)=sqrt(5)m`