Question 1140020
Find the height of a building if the angle of elevation to the top of the building changes from {{{18}}}° to {{{37}}}° as an observer moves a distance of {{{80 }}}feet toward the building.


 observer moves a distance of {{{80}}} (from original point) feet toward the building, and rest of a distance to building is {{{x}}} 

observe triangle with legs {{{x}}} and {{{h}}}(height)

=>{{{tan ( 37)=h/x}}}

{{{x=h/tan ( 37)}}}


total distance from the building is
 
{{{x+80=h/tan ( 37)+80}}}


then triangle with legs {{{x+80}}} and {{{h}}} 


{{{tan ( 18)=h/(h/tan ( 37)+80)}}}


{{{(h/tan ( 37)+80)*tan ( 18)=h}}}


{{{((h*tan ( 18))/tan ( 37))+80*tan ( 18)=h}}}


{{{h(tan ( 18)/tan ( 37))+80*tan ( 18)=h}}}


{{{h(tan ( 18)/tan ( 37))-h=-80*tan ( 18)}}}


{{{h(tan ( 18)/tan ( 37)-1)=-80*tan ( 18)}}}


{{{h=(-80*tan ( 18))/(tan ( 18)/tan ( 37)-1)}}}


{{{h=45.7ft}}}-> height of the building


then

{{{x=45.7/tan ( 37)=60.65ft}}}

ad total distance is {{{80+60.65=140.65ft}}}

check angles:

{{{tan^-1(45.7/60.65)=37}}}°
{{{tan^-1(45.7/140.65)=18}}}°...which confirms solution