From where kiana is standing, she must look up at a 50° angle to see the top of a building. if she backs up 50 feet, she will need to look up at a 40° angle to see the top of the same building. approximately how tall is the building?

For this case, what we must do is solve the following system of equations:
 tan (50) = h / x
 tan (40) = h / (x + 50)
 Solving the system we have:
 (x + 50) * tan (40) = h
 (x) * tan (50) = h
 Matching:
 (x + 50) * tan (40) = (x) * tan (50)
 Rewriting:
 x (tan (50) - tan (40)) = 50 * tan (40)
 x = 50 * tan (40) / (tan (50) - tan (40))
 x = 118.9692621
 Substituting:
 h = (x) * tan (50)
 h = (118.9692621) * tan (50)
 h = 141.7820455
 
 The height of the building is:
 h = 141.7820455 ft