Question

The angles of elevation of the top of a building and the top of the chimney on the roof of the building from a point on the ground are x and ${45}^0$ respectively. The height of building is h metre.Then the height of the chimney, in metre is:

• A. $h\cot x+h$
• B. $h\cot x-h$
• C. $h\tan x-h$
• D. $h\tan x+h$

Answer 1

Answer: (C)

$h\tan x-h$

Height of building ,$AD=h$

Let height of chimney, $CD=z$

In $△ABC$

$\tan x=\frac{AC}{AB}$

$\frac{1}{\cot x}=\frac{AC}{AB}$

$AB=AC\cot x=(h+z)\cot x.....\left(i\right)$

In $△ABD$

$\tan 45=\frac{AD}{AB}$

$\frac{1}{\cot 45}=\frac{AD}{AB}$

$AB=AD\cot 45=h.....\left(ii\right)$

From (i) and (ii)

$\Rightarrow (h+z)\cot x=h$

$\Rightarrow h\cot x+z\cot x=h$

$\Rightarrow z=\frac{h(1-\cot x)}{\cot x}$

$\Rightarrow z=\frac{h(1-\frac{h}{\tan x})}{\frac{1}{\tan x}}$

$\Rightarrow z=h(\tan x-1)$

$\Rightarrow z=h\tan x-h$

Hence height of chimney is $(h\tan x-h)$