Question

A suspension bridge has two main towers of equal height. A visitor on a tour ship approaching the bridge estimates that the angle of elevation to one of the towers is 19°. After sailing 419 ft closer he estimates the angle of elevation to the same tower to be 56°. Approximate the height of the tower.

Answer 1

189.917 ft

Explanation:

Let the height be h and the original distance from the tower be x.

`\tan 56^(\circ)=(h)/(x-419) \implies h=(x-419) \tan 56^(\circ) \\ \\ \tan 19^(\circ)=(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=(419\tan 56^(\circ))/(\tan 56^(\circ)-\tan 19^(\circ)) \\ \\ \implies h=(419\tan 56^(\circ)\tan 19^(\circ))/(\tan 56^(\circ)-\tan 19^(\circ)) \approx 187.917`