Question

From the top of a tree on one side of a street the angles of elevation and depression of the top and foot of a tower on the opposite side are respectively found to be $\alpha$ and $\beta$. If $h$ is the height of the tree, then the height of the tower is:

• A. $\frac{h\sin (\alpha +\beta )}{\cos \alpha \sin \beta }$
• B. $\frac{h\sin (\alpha +\beta )}{\sin \alpha \cos \beta }$
• C. $\frac{h\cos (\alpha -\beta )}{\cos \alpha \cos \beta }$
• D. $\frac{h\cos (\alpha +\beta )}{\cos \alpha \cos \beta }$

Answer 1

Answer: (A)

$\frac{h\sin (\alpha +\beta )}{\cos \alpha \sin \beta }$

Let $h_t+h=H$ be the total height of tower.
Let $x$ be the distance between tower and tree.
$\tan \beta =\frac{h}{x}$
$\tan \alpha =\frac{h_t}{x}$
Now,
$\tan \alpha =\frac{h_t}{h\cot \beta }$
$h_t=h\frac{\tan \alpha }{\tan \beta }$
Total height of the tower,
$H=h+h\frac{\tan \alpha }{tan\beta }$
$=\frac{h\tan \alpha +h\tan \beta }{\tan \beta }$
Substituting for $\tan \alpha$ and $\tan \beta$
we get,
$H=h\left(\frac{\sin \beta \cos \alpha +\sin \alpha \cos \beta }{\sin \beta \cos \alpha }\right)$
Hence, option 'A' is correct.