Question 1198040
<pre>
{{{drawing(400,200,-100,10,-10,45,

line(-95.263733,0,0,0),
green(triangle(-95.263733,0,0,0,0,38.489046505),
triangle(-95.263733+43,0,0,0,0,38.489046505)),

locate(-95.263733,0,B), locate(-95.263733+43,0,A),
locate(-85,7,22^o),locate(-85+39,7,38^o),
locate(-1,0,G),locate(-1,42,T),locate(-73,0,46),locate(-25,0,x),
locate(.3,20,y) )}}}

A and B are the two points from which the given angles of elevation
are measured. G is the base of the tree, and T is the top of the tree.
x is the distance from A to the base of the tree G.

Using right triangle ATG,


{{{TG/AG=opposite/adjacent=y/x=tan(38^o)}}}

Using right triangle BTG,


{{{TG/BG=opposite/adjacent=TG/(BA+AG)=y/(46+x)=tan(22^o)}}}

So the system to solve is

{{{system(y/(x+46)=tan(22^o),y/x=tan(38^o))}}}

which the other tutor gave.

Solving is easier if you let {{{tan(38^o)=u}}} and {{{tan(22^o)=v}}}

{{{system(y/(x+46)=v,y/x=u)}}}

Cross multiply:

{{{system(y=v(x+46),y=ux)}}}

{{{system(y=vx+46v,y=ux)}}}

Solve the 2nd equation for x

{{{x=y/u}}}

Substitute that for x in the 1st equation:

{{{y=v(y/u)+46v}}}

Multiply through by u

{{{uy=vy+46uv}}}

Get the y terms on the left:

{{{uy-vy=46uv}}}

Factor out y:

{{{y(u-v)=46uv}}}

{{{y=(46uv)/(u-v)}}}

Substituting tan(38<sup>o</sup>) for u and tan(22<sup>o</sup>) for v

{{{y=matrix(1,2,38.48904651,feet)}}}

Edwin</pre>