At the Foot of a Mountain, the Elevation of It Summit is 45°; After Ascending 1000 M Towards the Mountain up a Slope of 30° Inclination, the Elevation is Found to Be 60°. - Mathematics

At the foot of a mountain, the elevation of it summit is 45°; after ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain.

Solution

Suppose, AB is a mountain of height t + x.

$\text{ In } \bigtriangleup DFC,$
$\sin30° = \frac{x}{1000}$
$\Rightarrow x = 1000 \times \left( \frac{1}{2} \right) = 500 m$
$\text{ and }$
$\tan30° = \frac{x}{y}$
$\Rightarrow y = \frac{x}{\tan30°} = 500\sqrt{3}$
$\text{ In } ∆ ABC,$
$\tan45°= \frac{t + x}{y + z}$
$\Rightarrow t + x = y + z . . . \left( 1 \right)$
$\text{ In } ∆ ADE,$
$\tan60° = \frac{t}{z}$
$\Rightarrow t = \sqrt{3}z . . . \left( 2 \right)$
$\text{ From } \left( 1 \right) \text{ and } \left( 2 \right), \text{ we have }$
$\sqrt{3}z + x = y + z$
$\Rightarrow z\left( \sqrt{3} - 1 \right) = 500\left( \sqrt{3} - 1 \right)$
$\Rightarrow z = 500 m$
$\therefore t = \sqrt{3}z = 500\sqrt{3}$
Hence, height of the mountain =$t + x = 500\sqrt{3} + 500 = 500\left( \sqrt{3} + 1 \right) m$

Concept: Sine and Cosine Formulae and Their Applications
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 10 Sine and cosine formulae and their applications
Exercise 10.1 | Q 29 | Page 14