A bridge is situted at right-angle to the bank of a river. If one moves away a certain distance from the bridge along this side of the river, the other of the bridge is seen at an angle of `45^(@)` and if someone moves a further distance of 400 metres in the same direction, the other end is seen at an angle of `30^(@)`. Find the length of the bridge.

Explanation

Let the length of the bridge be AB. At the point bridge C the angle of elevation of the end A of the bridge is `45^(@)`. At the point D, 400 metres away from C, the angle of elevation of the end A is `30^(@)`.
As per question, CD = 400 metres.
`angleACB = 45^(@) and angleADB = 30^(@)`.
Now, from the right-angled triangle ABC, we get
`tan 45^(@) = (AB)/(BC)` [ by defintion ]
or, `1 = (AB)/(BC) " " `or, BC = AB.............(1)
Again, from the right-angled triangle ABD we get,
`tan 30^(@) = (AB)/(BD)` [ by definition]
or, `(1)/(sqrt(3))=(AB)/(BD) " "or, " "sqrt(3)AB = BD " "or. " "sqrt(3)AB = BC + CD`
or, `sqrt(3)AB = AB + 400` [ from (1) ]
or, `sqrt(3)AB-AB = 400 " "or, " "AB(sqrt(3)-1) = 400 " "or, " "AB = (400)/(sqrt(3)-1)`
or, `AB = (400(sqrt(3)-1))/((sqrt(3))^(2)-(1)^(2)) " " AB = (400(sqrt(3)+1))/(2) " "or, " "AB = 200(sqrt(3) + 1)`
Hence the length of the bridge = `200(sqrt(3) + 1)` metres.