Question

A pole of height 10 m cast the shadow of length 12 m. Find the length of the pole and cast a shadow of length 15 m at the same time.

Answer 1

Hint: The formula for writing tangent of an angle is
\[\tan \theta =\dfrac{Perpendicular}{\text{Base}}\] .

Complete step-by-step answer:
As mentioned in the question, the figure would look like the below picture

By inspecting the figure, we have

The pole of height PQ which is given in the question is given as =10m
The shadow length of the pole QR as mentioned in the question is given as =12m
At the same time, what the question demands is given as follows
The pole of height PQ is unknown so let it be x.
The shadow length of the pole QR as mentioned in the question is given as
=15m
In ΔPQR, we can use the tangent formula as mentioned in the hint to get the following
\[\begin{array}{*{35}{l}}
   tan\theta =\ \dfrac{10}{12}~~~~~~\ldots \ldots ~\text{ }\left( 1 \right) \\
   {} \\
   {} \\
\end{array}\]
Similarly, for the second part we can write as
\[tan\theta =\dfrac{x~}{15}~~\text{ }\ldots \ldots .~\text{ }\left( 2 \right)\]
As angle is equal for same timing,
Hence, by equating equation (1) and (2), we get
\[\begin{align}
  & \dfrac{10}{12}=\dfrac{x}{15} \\
 & x=\dfrac{15\times 10}{12} \\
 & x=12.5\ cm \\
\end{align}\]
Hence, the height of the pole is 12.5 cm.

Note: The figure in this question is very tricky and is difficult to visualize it at first. Hence, the students can make an error while drawing the figure and then end up making a mistake and they would get to the correct solution.