Question 1196263
Let h = height of the hut

Expressing given problem to systems of equations

(1)  Tan 55 = h/(15-x) = 1.428

(2)  Tan 45 = h/x = 1 
          x = h

Substitute x = h in equation (1)
h/(15-h) = 1.428
       h = 1.428(15-h)
       h = 21.422 - 1.428h
  2.428h = 21.422
       h = 8.823 

Answer
The height of the hut is 8.823 meters.