A woman stands at a horizontal distance \"x\" from a mountain and measures the angel of elevation of the mountaintop above the horizontal as \"Θ.\" After walking a distance \"d\" closer to the mountain on ground level, she finds the angel to be \"Φ.\" Find a general equation for the height \"y\" of the mountain in terms of \"d\

we are given initially with angle of elevation of Θ and distance x. The height,y is equal to x tan Θ. when the woman stands closer, the angle changes to Φ and the distance in between becomes x-d. Hence the height is also expressed as (x-d) tan Φ. The height then is equal to x tan Θ - (x-d) tan Φ
or x tan Θ - x tan Φ + d tan Φ

Answer Link

johanrusli johanrusli · Answer 2 · 2019-07-14T15:54:56+02:00

A general equation for the height y of the mountain in terms of d is

y = ( d tan Ф tan Θ ) / ( tan Ф - tan Θ )

Further explanation

First, let us recall the following formula about trigonometry.

Suppose the ΔABC is a right triangle at A, then:

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

Let us now tackle the problem!

The problem can be seen as shown in the picture in the attachment.

Look at ΔABC :

[tex]\tan \Theta = \frac{opposite}{adjacent}[/tex]

[tex]\tan \Theta = \frac{AC}{AB}[/tex]

[tex]\tan \Theta = \frac{y}{x}[/tex]

[tex]x = \frac{y}{\tan \Theta}[/tex] → Equation 1

Look at ΔADC :

[tex]\tan \Phi = \frac{opposite}{adjacent}[/tex]

[tex]\tan \Phi = \frac{AC}{AD}[/tex]

[tex]\tan \Phi = \frac{y}{x - d}[/tex]

[tex]y = (x - d)\tan \Phi[/tex]

[tex]y = (\frac{y}{\tan \Theta} - d)\tan \Phi[/tex] ← Equation 1

[tex]y = \frac{y ~ \tan \Phi}{\tan \Theta} - d ~ \tan \Phi[/tex]

[tex]\frac{y ~ \tan \Phi}{\tan \Theta} - y = d ~ \tan \Phi[/tex]

[tex]y ( \frac{\tan \Phi}{\tan \Theta} - 1) = d ~ \tan \Phi[/tex]

[tex]y ( \frac{\tan \Phi - \tan \Theta}{\tan \Theta} ) = d ~ \tan \Phi[/tex]

[tex]\large {\boxed {y = \frac{d ~ \tan \Phi ~ \tan \Theta}{\tan \Phi - \tan \Theta}} }[/tex]

Learn more

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Trigonometry Formula : https://brainly.com/question/12668178

Answer details

Grade: High School

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse