Question 1207217
.
A person standing 400 ft from the base of a mountain measures the angle of elevation 
from the ground to the top of the mountain to be 25°. The person then walks 500 ft 
straight back and measures the angle of elevation to now be 20°. How tall is the mountain?
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        The solution in the post by  @Theo is  INCORRECT.

        400  feet that  Theo marked in his plot,  represents, actually, the base of the mountain,  which is  NOT  GIVEN  in the problem.

        Actually, this data of  400 feet plays small role in this problem  (if any)  and only confuses a reader.

        The data which really  PLAYS  a role is  500  feet,  the distance between the two positions.


        I came to bring a correct solution.



<pre>
Let L be the distance (on the ground level, i.e. horizontally) from the first position to the 
top of the mountain. So, we can write

    {{{x/L}}} = tan(25°),  or  x = tan(25°)*L.   (1)

where x is the height of the mountain.


Then the distance (on the ground level, i.e. horizontally) from the second position to the 
top of the mountain is  (L+500) feet. So, we can write

    {{{x/(L+500)}}} = tan(20°),  or x = tan(20°)*(L+500).  (2)


In equations (1) and (2) left sides are identical, so their right side are equal

    tan(25°)*L = tan(20°)*(L+500).


Simplify and find L

    tan(25°)*L = tan(20°)*L + 500*tan(20°),

    tan(25°) - tan(20°)*L = 500*tan(20°),

    L = {{{(500*tan(20^o))/(tan(25^o)-tan(20^o))}}}.


Now substitute it into equation (1) and find x

    x = {{{(500*tan(20^0)*tan(25^o))/(tan(25^o)-tan(20^o))}}} = {{{(500*0.466307658*0.363970234)/(0.466307658-0.363970234)}}} = 829.228 ft.


Round to the closest nearest whole foot and get the


<U>ANSWER</U>.  The height of the mountain is about 829 feet.
</pre>

Solved (correctly).



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To see many other solved similar and different problems of the type &nbsp;" find the height ",
look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Find-the-height.lesson>Find the height</A> 

in this site &nbsp;(free of charge).


Learn the subject from there.