Question

An engineer determines that the angle of elevation from her position to the top of a tower is 32°. She measures the angle of elevation again from a point 50 m closer to the tower and finds it to be 52°. Find the height of the tower

Answer 1

Given :

The angle of elevation from her position to the top of a tower is 32°.

The angle of elevation again from a point 50 m closer to the tower and finds it to be 52°.

The following figure represents the given situation :

Let the height of the tower h

And the distance between the tower and the second point = x

So, from the larger triangle :

`\begin{gathered} \tan 32=(h)/(x+50) \\ \\ h=(x+50)\cdot\tan 32 \end{gathered}`

From the smaller triangle :

`\begin{gathered} \tan 52=(h)/(x) \\ \\ h=x\cdot\tan 52 \end{gathered}`

So,

`(x+50)\cdot\tan 32=x\tan 52`

solve for x:

`\begin{gathered} x\cdot\tan 32+50\tan 32=x\cdot\tan 52 \\ 50\cdot\tan 32=x\cdot\tan 52-x\cdot\tan 32 \\ 50\cdot\tan 32=x\cdot(\tan 52-\tan 32) \\ \\ x=(50\cdot\tan 32)/(\tan 52-\tan 32)\approx47.7 \end{gathered}`

so, the height h will be :

`h=x\cdot\tan 52=47.7\cdot\tan 52\approx61m`

So, the height of the tower​ = 61 m