Question

Determine the height of a mountain if the elevation of its top at an unknown distance from the base is ${30}^{\circ }$ and at a distance 10 km further off from the mountain, along the same line , the angle of elevation is ${15}^{\circ }$ . (Use tan${15}^{\circ }$ = 0.27)

Answer 1

Let AB be the mountain of height h kilometer . Let C be point at a distance of x km . from the base of the mountain such that the angle of elevation of the top at C is ${30}^{\circ }$ . Let D be a point at a distance of 10 km from C such that the angle of elevation at D is of${15}^{\circ }$
In $△$ CAB , we have
$tan{30}^{\circ }=\frac{AB}{AC}$
$\Rightarrow \frac{1}{\sqrt{3}}=\frac{h}{x}$
$\Rightarrow x=\sqrt{3h}$
In $△$ DAB we have
$tan{15}^{\circ }=\frac{AB}{AD}$
$\Rightarrow 0.27=\frac{h}{x+10}$
$\Rightarrow$ (0.27) (x + 10) = h
substituting x = $\sqrt{3h}$ obtained from equation (i) in equation (ii) we get
0.27 ( $\sqrt{3h}$ + 10) = h
$\Rightarrow 0.27\times 10=h-0.27\times \sqrt{3h}$
$\Rightarrow$ h (1 - 0.27 $\times \sqrt{3})$ = 2.7
$\Rightarrow$ h (1 - 0.46 ) = 2 . 7
$\Rightarrow h=\frac{2.7}{0.54}$ = 5
Hence , the height of the mountains is 5 km