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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. 

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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
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