Question

From the top of a hill, the angles of depression of two consecutive kilometer stones due east are found to be ${30}^o$ and ${45}^o$. Find the height of the hill

• A. $1.365$km
• B. $1.5$km
• C. $1.7$km
• D. $1.1$km

Answer 1

The correct option is A $1.365$km

Let the distance between the nearer kilometre stone and the hill be '$x$' km.

So, the distance between the farther kilometre stone and the hill is '$1+x$' km since both are on the same side of the hill.

In triangle APB,

$\tan {45}^0=\frac{h}{x}$

$\Rightarrow$$1=\frac{h}{x}$

$\Rightarrow$$h=x$

In triangle AQB,

$\tan {30}^0=\frac{h}{1+x}$

$\Rightarrow$$\frac{1}{\sqrt{3}}=\frac{h}{1+x}$

$\Rightarrow$$1+x=\sqrt{3}h$

From equation 1,

$1+h=\sqrt{3}h\Rightarrow$$1=\sqrt{3}h-h$

$\Rightarrow$$h=\frac{1}{\sqrt{3}-1}$

$\Rightarrow$$h=1.365km$

Hence option A is correct.