Question

Determine the height of a mountain if the elevation of its top at an unknown distance from the base is 30° and at a distance 10km further off from the mountain,along the same line, the angle of elevation is 15°.(Usetan15°=0.27)​

Answer 1

EXPLANATION.

$\sf : \implies \: ab \: \: be \: the \: mountain \: of \: height \: = \: h \: m \\ \\ \sf : \implies \: in \: \triangle \: cab \: \\ \\ \sf : \implies \: \tan( \theta) = \frac{perpendicular}{base} \\ \\ \sf : \implies \: \tan(30 \degree) = \frac{ab}{bc} \\ \\ \sf : \implies \: \frac{1}{ \sqrt{3} } = \frac{h}{x} \\ \\ \sf : \implies \: x \: = \sqrt{3}h$

$\sf : \implies \: in \: \triangle \: dab \: \\ \\ \sf : \implies \: \tan(15 \degree) = \frac{ab}{ad} \\ \\ \sf : \implies \: 0.27 = \frac{h}{x + 10} \\ \\ \sf : \implies \: (0.27)(x + 10) = h$

$\sf : \implies \: put \: the \: value \: of \: x \: = \sqrt{3} h \: in \: second \: equation \\ \\ \sf : \implies \: 0.27( \sqrt{3}h \: + 10) = h \\ \\ \sf : \implies \: 0.27 \times \sqrt{3}h + 0.27 \times 10 = h \\ \\ \sf : \implies \: 0.27 \times 10 = h \: - 0.27 \times \sqrt{3} h \\ \\ \sf : \implies \: 2.7 = h \: (1 - 0.27 \times \sqrt{3} ) \\ \\ \sf : \implies \: h \: = \frac{2.7}{0.54} = 5$

$\sf : \implies \: \orange{{ \underline{height \: of \: mountain \: = \: 5 \: km}}}$

Answer 2

Answer:

⭐ Solutions :-

Let AB be the mountain of the height (h) kilometres. 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°. Let D be a point at a distance of 10 km from C such that the angle of elevation at D is of 15°.

In ∆ CAB we have ,

tan30° = $\dfrac{AB}{AC}$

$\implies$ $\dfrac{1}{√3}$ = $\dfrac{h}{x}$

$\implies$ x = √3h

In ∆ DAB we have ,

tan15° = $\dfrac{AB}{AD}$

$\implies$ 0.27 = $\dfrac{h}{x + 10}$

$\implies$ (0.27)(x + 10) = h

Substituting x = √3h obtained from equation (1) in equation (2) we get 0.27 (√3h + 10) = h

=> 0.27 × 10 = h - 0.27 × √3h

=> h (1 - 0.27 × √3) = 2.7

=> h (1 - 0.46 ) = 2.7

=> h = $\dfrac{2.7}{0.54}$

=> h = 5

Step-by-step explanation:

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