The angle of elevation of the top A of a building from a point C due south of it is 25 degrees. At a second point D, which is 160 metres due west of C, the angle of elevation of the top of the building is 20 degrees. Point B is the bottom of the vertical building and on the same horizontal plane as D and C.

- Find the height of AB of the building to the nearest metre.

Guest Jun 19, 2014

#1**+14 **

Best Answer

Let C be the point (0,0,0)

Then D is (-160,0,0)

B is (0,y,0)

A is (0,y,z)

From what we are given

$$\tan(25deg) = \dfrac z y$$

$$\tan(20deg) = \dfrac{z}{\sqrt{160^2+y^2}}$$

solving this we get

z=93.16m which is the the length of AB, i.e. the height of the building

Rom
Jun 19, 2014

#2**+5 **

Using the information in the first part of the problem, let d be the distance from C to B and h be the height of the building. So we have,

tan(25) = h/d Solving for h we have, h = d*tan(25)

And, since we have a right triangle with two legs d and 160, the distnce that D is from B = √(160^{2} + d^{2})

So we have

tan(20) = (h)/√(160^{2} + d^{2}) = (d*tan(25)) / √(160^{2} + d^{2})

Simpifying, we have

tan(20)/ tan(25) = d/√(160^{2} + d^{2}) and we can write

tan(25)/tan(20) = √(160^{2} + d^{2}) /d Square both sides

[tan(25)/tan(20)]^{2} = (160^{2} + d^{2})/d^{2} and the right side = 160/d^{2} + 1 Subtract 1 from both sides

([tan(25)/tan(20)]^{2} - 1) = 160^{2}/d^{2} and we can write

d^{2} = 160^{2} / ([tan(25)/tan(20)]^{2} - 1) Simplifying this, we have

d^{2} = 160^{2} / (.6413959572703699169) Now....take the square root of both sides

d = 199.7822378461935026188794268 m

And using .... h = d*tan(25) ....we have....

h = (199.7822378461935026188794268)* tan(25) ≈ 93.159 m or just 93 m

I think that's it.....I haven't worked one like this before....thanks for submitting it....I hope I haven't made any grevious mistakes !! (P.S. ....thanks to * alan *for catching my previous error.....I went back and corrected it so that now, it appears that I'm smarter than I might really be....also..

CPhill
Jun 19, 2014