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Tony is standing at sea level. From his location, the angle of elevation of the top of Blue...

Question:

Tony is standing at sea level. From his location, the angle of elevation of the top of Blue Mountain is {eq}23^{\circ} {/eq}. Staying at sea level, he walks 200 yards toward the mountain. The angle of elevation of the top is now {eq}27^{\circ} {/eq}.

Find the height of Blue Mountain.

Angle Of Elevation:

Angle of Elevation is defined as the angle formed between the line of sight of the observer and the horizontal. For example when an observer sees an object which is a height from ground level then the angle which the observer's eye makes with the horizontal is known as angle of elevation.

Answer and Explanation: 1

Here, we try to understand the question with respect to the given figure:

In the figure we have divided the question into two triangles i.e., Triangle ABC and Triangle ABD

{eq}In\; \Delta ABC\\ Let\; AB\;=\;height\; of\;the\;Blue\;mountain\;=\;x\\ BC\; be\;the\;distance\;the\;man\;and\;the\;mountain\;=\;y\\ \tan\theta = \frac{perpendicular}{base}\\ \tan\theta = \frac{AB}{BC}\\ \tan\left ( 27^{\circ} \right ) = \frac{x}{y}\\ y = \frac{x}{\tan\left ( 27^{\circ} \right )} \\ y = 1.96x\\ In\;\Delta ABD\\ AB\;=\;x\\ BD\;=\;200\;+\;y\\ \tan\theta = \frac{AB}{BD}\\ \tan\theta = \frac{perpendicular}{base}\\ \tan\left ( 23^{\circ} \right ) = \frac{x}{200+y}\\ = \frac{x}{200+1.96x}\\ 0.4245 = \frac{x}{200+y}\\ 0.4245*\left ( 200+1.96x \right ) = x\\ 84.9 + 0.83202x = x\\ x - 0.83202x = 84.9\\ 0.16798x = 84.9\\ \boxed{x = 505.417\approx 505.42} {/eq}

Therefore, the height of the Blue Mountain is 505.42 yards.


Learn more about this topic:

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What is Trigonometry? - Functions, Formulas & Applications

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Chapter 22 / Lesson 11
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Learn what trigonometry is and what trigonometric functions are. Understand the examples of how to use each function, as well as know the instances when it is useful to use trigonometry.


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