Question

$A$ balloon is observed simultaneously from three points $A,B,C$ due west of it on a horizontal line passing directly underneath it. lf the angular elevations at $B$ and $C$ are respectively twice and thrice that at$A$and if $AB=220$ metres and $BC=100$ metres, then the height of the balloon from the ground is

• A. $167$ mt
• B. $176$ mt
• C. $184$ mt
• D. $192$ mt

Answer 1

The correct option is B $176$ mt

$h=dtan3\alpha$
$h=(d+100)tan2\alpha$
$h=(320+d)tan\alpha$
$h=220sin2\alpha$
$=440sin\alpha cos\alpha$
$\frac{100}{sin\alpha }=\frac{PB}{sin3\alpha }=\frac{PC}{sin2\alpha }$
$in△PCB$
$\frac{320}{sin2\alpha }=\frac{PC}{sin\alpha }=\frac{AP}{sin3\alpha }$
in $△$ PCA
$\therefore PC=\sqrt{32000}$
$\frac{2sin\alpha cos\alpha }{sin\alpha }=\frac{320}{\sqrt{32000}}$
$cos\alpha =\frac{160}{\sqrt{32000}}$
$sin\alpha =\sqrt{0.2}$
$\therefore h=440\times \sqrt{0.2}\times \frac{160}{\sqrt{32000}}$
$\therefore h=176$