Question

The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-story building are 30 and 45 degrees, respectively. What is the height of the multi-story building and the distance between them?

Answer 1

The height of the multi-story building is `18.93` m and
distance between the two buildings is also `18.93` m.

Explanation:

Angle of depression of top of the `8` m tall building

from the top of the `h` m multi-story building is `theta_1=30^0`

and angle of depression of bottom of the `8` m tall building

from the top of the `h` m multi-story building is `theta_2=45^0`

Let the difference of their height `x=h-8 :. tan30 = x/d` or

`d=x*sqrt3; (1)` , where `d` is the horizontal distance between

two buildings. Also `h/d =tan45 or (x+8)/d= tan45` or

`(x+8)=d; (2)` From equation (1) and equation (2) we get

`x+8= sqrt3*x or x(sqrt3-1) =8 or x= 8/(sqrt3-1)~~10.93` m

`:. h ~~ 10.93+8= 18.93` m and `d= x+8= 18.93` m

The height of the multi-story building is `18.93` m and

distance between the two buildings is also `18.93` m [Ans]

Answer 2

Height of the multistory building = Distance between the

buildings and the value is `color(green)(18.9282)` `color(red)m`

Explanation:

Given `/_(BAC) = 30^0, /_(BAD) = 45^0, CD = 8 m`

To find height of the multistory building AF and distance ED = AB

`tan (BAC) = tan 30^0 = (BC) / (AB)` Eqn 1

`tan (BAD) = tan 45^0 = (BD) / (AB)` Eqn 2

Since `tan 45 = 1, BD = AB`

Replacing AB by BD in Eqn 1,

`BC = AB * tan 30 = BD * tan 30 = (BC + CD) * (1/sqrt3)`

`BC * sqrt 3 = BC + 8`

`BC = 8 / (sqrt3 - 1) ~~ 8 / 0.732 = 10.9282`

Therefore height of the multistory building and distance between the two buildings is
`AB = BD = BC + CD = 10.9282 + 8 = color(green)(18.9282)` `color(red)m`