#### Question

From a point on a bridge across a river, the angles of depression of the banks on opposite side of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, find the width of the river.

#### Solution

Let BD be the width of the river. And the angle of depression of the bank on opposite side of the river is 30° and 45° respectively. It is given that AC = 30 m.

Let BC = x and CD = y. And ∠ABC = 30°, ∠ADC = 45°.

Here we have to find the width of the river.

We have the following figure

So we use trigonometric ratios.

In a triangle ABC

`=> tan 30° = (AC)/(BC)`

`=> 1/sqrt3 = 30/x`

`=> x = 30sqrt3`

Again in a triangle ADC

`=> tan 45° = (AC)/(CD)`

`=> 1 = 30/y`

`=> y = 30`

So width of river is

`x + y = 30sqrt3`

`x + y = 30(sqrt3 + 1)`

Hence the width of river is `(30(sqrt3 + 1)`m