Question

An observer, 1.5m tall, is 28.5m away from a tower 30m high. Determine the angle of elevation of the top of the tower from his eye.

Answer 1

Hint: First we will draw the diagram for a better understanding, then we will use the trigonometric formula like $\tan \theta =\dfrac{height}{base}$ to find the angle of elevation and after that we will substitute all the given values.

Complete step-by-step answer:
Let’s first draw the required diagram,

We need to find the value of $\theta $ , as it is the angle of elevation of the top of the tower from the man’s eye.
We have drawn a line DE which is parallel to BC.
Therefore we get,
 DE = BC
Hence, it is given that BC = 28.5.
And we know that DE is parallel to BC.
Therefore from this we get,
DE = 28.5
Now the height of the tower AB = 30m and the height of the man DC = 1.5m,
We know that BE is parallel to DC,
Therefore we get,
BE = DC = 1.5m
Now from the diagram we can see that,
AB = AE + BE
Now substituting the values of AB = 30m and BE = 1.5m we get,
AE = 30 – 1.5
AE = 28.5m
Now in the triangle ADE, we need to find the value of $\theta $,
$\tan \theta =\dfrac{height}{base}$
Now substituting the values of height(AE) = 28.5 and base(DE) = 28.5 we get,
$\begin{align}
  & \tan \theta =\dfrac{28.5}{28.5} \\
 & \tan \theta =1 \\
 & \Rightarrow \theta =\dfrac{\pi }{4} \\
\end{align}$
Hence, the angle of elevation is 45 degrees.

Note: The formula that we have used must be kept in mind, and the meaning of angle of elevation must be clear to avoid any mistake while solving the question. One can also directly see that if in a right angle triangle two sides are the same then the angle is 45 degree, so from this we can directly write the angle of elevation is 45 degree.