#### Question

The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30° and 45° respectively. Find the height of the multistoried building and the distance between the two buildings.

#### Solution

Let *AD* be the multistoried building of height hm. And the angle of depression of the top and bottom are 30° and 45°. We assume that *BE* = 8, *CD* = 8 and *BC* = *x*, *ED = x* and *AC* = *h* − 8. Here we have to find height and distance of the building.

We use trigonometric ratio.

In ΔAED,

`=> tan E = (AD)/(DE)`

`=> tan 45^@= (AD)/(DE)`

`=> 1 = h/x`

=> x = h

Again in Δ ABC

`=> tan B = (AC)/(BC)`

`=> tan 30^@ = (h - 8)/x`

`=> 1/sqrt3 = (h - 8)/x`

`=> hsqrt3 - 8sqrt3 = x`

`> hsqrt3 - 8sqrt3 = h`

`=> h(sqrt3 - 1) = 8sqrt3`

`=> h = (8sqrt3)/(sqrt3 - 1) xx (sqrt3 + 1)/(sqrt3 + 1)`

`=> h = (24 + 8sqrt3)/2`

`=> h = (4(3 + sqrt3))`

And

`=> x = 4(3 + sqrt3)`

Hence the required height is `4(3 + sqrt3)` meter and distance is `4(3 + sqrt3)` meter