Question

The angles of elevation of the top of a building from the top and bottom of a tree are $x$ and $y$ respectively. If the height of the tree is $h$ meter then, in meter, the height of the building is:

• A. $\frac{hcotx}{cotx+coty}$
• B. $\frac{hcoty}{cotx+coty}$
• C. $\frac{hcotx}{cotx-coty}$
• D. $\frac{hcoty}{cotx-coty}$

Answer 1

Answer: (A)

$\frac{hcotx}{cotx+coty}$

In $△AED$

$\tan x=\frac{DE}{AE}\therefore \tan x=\frac{H-h}{AE}$------- (1)

$AE=\frac{H-h}{\tan x}$------(A)

In $△DBC$

$\tan y=\frac{DC}{BC}\therefore \tan y=\frac{H}{AE}$------- (2)

$AE=\frac{H}{\tan y}$------ (B)

Comparing $A$ and $B$

$\frac{H-h}{\tan x}=\frac{H}{\tan y}$

$(H-h)\tan y=H.\tan x$

$H\tan y-H\tan x=h\tan y$

$H=\frac{h\tan y}{\tan y-\tan x}$
$=\frac{\frac{h}{\cot y}}{\frac{1}{\cot y}-\frac{1}{\cot x}}$
$=\frac{\frac{\frac{h}{\cot y}}{\cot x-\cot y}}{\cot x.\cot y}$
$=\frac{h\cot x}{\cot x-\cot y}$