The data the book has supplied in its diagram is geometrically impossible. In a right-triangle, you can't have a hypotenuse (14.6 m) shorter than one of the other sides (14.79 m). I say.
It's geometrically impossible for the angle at E to be a right angle if DG = 14.6 and triangle EFG is as given ..... Alternatively, DG = 14.6 is geometrically impossible if the angle at E is a right angle and triangle EFG is as given ......
One of the following is clearly wrong (perhaps a misprint): 11.7 m, 14.6 m, 52.3 degrees.
PS: topsquark, x = 14.79 is correct ...... I'd almost guess that you were in radian mode by mistake except you'd get 13.08, not 13.9 ...... Maybe radian mode and bad light ......