Respuesta :
Given
While sailing a boat offshore, Bobby sees a lighthouse and calculates that
the angle of elevation to the top of the lighthouse is 3°.
When she sails her boat 700 m closer to the lighthouse, she finds that the angle of elevation is now 5°.
To find:
How tall, to the nearest tenth of a meter, is the lighthouse?
Explanation:
It is given that,
While sailing a boat offshore, Bobby sees a lighthouse and calculates that
the angle of elevation to the top of the lighthouse is 3°.
When she sails her boat 700 m closer to the lighthouse, she finds that the angle of elevation is now 5°.
That implies,
[tex]\tan3\degree=\frac{y}{x+700}[/tex]Also,
[tex]\begin{gathered} \tan5\degree=\frac{y}{x} \\ y=x\tan5\degree \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} \tan3\degree=\frac{x\tan5\degree}{x+700} \\ (x+700)\tan3\degree=x\tan5\degree \\ x\tan3\degree+700\tan3\degree=x\tan5\degree \\ (\tan5\degree-\tan3\degree)x=700\tan3\degree \\ 0.0351x=36.6854 \\ x=1045.7m \end{gathered}[/tex]Then,
[tex]\begin{gathered} \tan5\degree=\frac{y}{1045.7} \\ y=1045.7\tan5\degree \\ y=91.5m \end{gathered}[/tex]Hence, the height of the light house is, 91.5m.
