A surveyor determines that the angle of elevation to the top of a building from a point on the ground is 31∘. he then moves back 57.1 feet and determines that the angle of elevation is 23.8∘. what is the height of the building?

Hello!The answer is:The height of the building is 94.29 feet.Why?To solve the problem, we need to consider that two triangles are formed depending on the distance where the surveyor is standing. We have a first triangle which its base will be called "x" and its height will be called "h" with an angle of elevation from a point on the ground, equal to 31°.  Also, we have a second triangle which base is equal to "x" + 57.1 feet, and a height "h", with an angle of elevation from a point on the ground, equal to 23.8°We need to use a trigonometric identity that establishes a relationship between the height and the base. So, using the tangent trigonometric identity, we have:[tex]Tan(\alpha)=\frac{Opposite}{Adjacent}[/tex]For the first triangle:[tex]Tan(31\°)=\frac{h}{x}[/tex][tex]0.60=\frac{h}{x}[/tex][tex]h=0.60*x=0.60x[/tex]For the second triangle:[tex]Tan(23.8\°)=\frac{h}{x+57.1feet}[/tex][tex]0.44=\frac{h}{x+57.1feet}[/tex][tex](0.44)*(x+57.1feet)=h[/tex][tex]h=0.44x+25.14feet[/tex]Then, making both equation equals, in order to find "x", we have:[tex]0.60x=0.44x+25.14feet[/tex][tex]0.60x-0.44x=25.14feet[/tex][tex]0.16x=25.14feet[/tex][tex]x=\frac{25.14feet}{0.16}=157.13feet[/tex]Then, we substituting "x" into any of the equations, we have:Substituting into the first equation:[tex]h=0.60*x\\\\h=0.60*(157.13feet)=94.29feet[/tex]Hence, the height of the building is 94.29 feet.Have a nice day!