<h2>
Hello!</h2>
The answer is:
The height of the building is 94.29 feet.
<h2>
Why?</h2>
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:
![Tan(\alpha)=\frac{Opposite}{Adjacent}](https://tex.z-dn.net/?f=Tan%28%5Calpha%29%3D%5Cfrac%7BOpposite%7D%7BAdjacent%7D)
For the first triangle:
![Tan(31\°)=\frac{h}{x}](https://tex.z-dn.net/?f=Tan%2831%5C%C2%B0%29%3D%5Cfrac%7Bh%7D%7Bx%7D)
![0.60=\frac{h}{x}](https://tex.z-dn.net/?f=0.60%3D%5Cfrac%7Bh%7D%7Bx%7D)
![h=0.60*x=0.60x](https://tex.z-dn.net/?f=h%3D0.60%2Ax%3D0.60x)
For the second triangle:
![Tan(23.8\°)=\frac{h}{x+57.1feet}](https://tex.z-dn.net/?f=Tan%2823.8%5C%C2%B0%29%3D%5Cfrac%7Bh%7D%7Bx%2B57.1feet%7D)
![0.44=\frac{h}{x+57.1feet}](https://tex.z-dn.net/?f=0.44%3D%5Cfrac%7Bh%7D%7Bx%2B57.1feet%7D)
![(0.44)*(x+57.1feet)=h](https://tex.z-dn.net/?f=%280.44%29%2A%28x%2B57.1feet%29%3Dh)
![h=0.44x+25.14feet](https://tex.z-dn.net/?f=h%3D0.44x%2B25.14feet)
Then, making both equation equals, in order to find "x", we have:
![0.60x=0.44x+25.14feet](https://tex.z-dn.net/?f=0.60x%3D0.44x%2B25.14feet)
![0.60x-0.44x=25.14feet](https://tex.z-dn.net/?f=0.60x-0.44x%3D25.14feet)
![0.16x=25.14feet](https://tex.z-dn.net/?f=0.16x%3D25.14feet)
![x=\frac{25.14feet}{0.16}=157.13feet](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B25.14feet%7D%7B0.16%7D%3D157.13feet)
Then, we substituting "x" into any of the equations, we have:
Substituting into the first equation:
![h=0.60*x\\\\h=0.60*(157.13feet)=94.29feet](https://tex.z-dn.net/?f=h%3D0.60%2Ax%5C%5C%5C%5Ch%3D0.60%2A%28157.13feet%29%3D94.29feet)
Hence, the height of the building is 94.29 feet.
Have a nice day!