The angles of elevation and depression of the top and the bottom of a tower from the top of a building, $60\ m$ high, are $30^{o}$ and $60^{o}$ respectively. Find the difference between the heights of the building and the tower and the distance between them.
Given: Height of the building$=60\ m$ and angles of elevation and depression of the top and the bottom of a tower from the top of a building, $30^{o}$ and $60^{o}$
To do: To find the difference between the heights of the building and the tower and the distance between them.
Solution:
Let AB be the building and CD is the tower, as shown in the fig,
In right $\vartriangle ABD$,
$tan60^{o}=\frac{AB}{BD}$
$\Rightarrow \sqrt{3} =\frac{60}{BD}$
$\Rightarrow BD=\frac{60}{\sqrt{3}}$
$\Rightarrow BD=20\sqrt{3} \ m$
In right $\vartriangle ACE$,
$tan30^{o}=\frac{CE}{AE}$
$\Rightarrow \frac{1}{\sqrt{3}} =\frac{CE}{BD}$
$\Rightarrow CE=\frac{BD}{\sqrt{3}}$
$\Rightarrow CE=\frac{20\sqrt{3}}{\sqrt{3}}$
$\Rightarrow CE=20\ m$
Height of the tower$=CE+ED=20+60=80\ m\ \ \ ( \because AB=ED=60\ m)$
Difference between the heights of the building and the tower$=80-60=20\ m$
Distance between the building and the tower$=BD=20\sqrt{3} \ m$.
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