Answer:
189.917 ft
Step-by-step explanation:
Let the height be h and the original distance from the tower be x.
[tex]\tan 56^{\circ}=\frac{h}{x-419} \implies h=(x-419) \tan 56^{\circ} \\ \\ \tan 19^{\circ}=\frac{h}{x} \implies h=x \tan 19^{\circ} \\ \\ \therefore (x-419)\tan 56^{\circ}=x\tan 19^{\circ} \\ \\ x\tan 56^{\circ}-419\tan 56^{\circ}=x\tan 19^{\circ} \\ \\ x(\tan 56^{\circ}-\tan 19^{\circ})=419\tan 56^{\circ} \\ \\ x=\frac{419\tan 56^{\circ}}{\tan 56^{\circ}-\tan 19^{\circ}} \\ \\ \implies h=\frac{419\tan 56^{\circ}\tan 19^{\circ}}{\tan 56^{\circ}-\tan 19^{\circ}} \approx 187.917[/tex]
