Answer:
1053.07 ft
Step-by-step explanation:
To find the horizontal distance [tex]\bold{\sf d }[/tex] from the ship to the lighthouse, we can use trigonometry and the concept of angle of elevation.
Given:
- Height of the lighthouse [tex]\bold{\sf h = 148 }[/tex] feet
- Angle of elevation [tex]\bold{\sf \theta = 8^\circ }[/tex]
The tangent of the angle of elevation [tex]\bold{\sf \theta }[/tex] is defined as the ratio of the opposite side (height of the lighthouse) to the adjacent side (horizontal distance [tex]\bold{\sf d }[/tex] from the ship to the lighthouse):
[tex] \large\boxed{\boxed{ \tan(\theta) = \sf \dfrac{Opposite}{Hypotenuse}}} [/tex]
So, we have
[tex]\sf \tan(\theta) = \dfrac{height}{distance} [/tex]
Substituting the given values:
[tex]\sf \tan(8^\circ) = \dfrac{148}{distance} [/tex]
To solve for [tex]\bold{\sf distance }[/tex], rearrange the equation:
[tex]\sf distance = \dfrac{148}{\tan(\theta)} [/tex]
[tex]\sf distance = \dfrac{148}{\tan(8^\circ)} [/tex]
[tex]\sf distance = \dfrac{148}{0.1405408347023} [/tex]
[tex]\sf distance \approx 1053.0747189128 [/tex]
[tex]\sf distance \approx 1053.07\textsf{feet (in nearest hundredth)}[/tex]
Therefore, the ship is approximately [tex]\bold{\sf \boxed{1053.07} }[/tex] feet from the lighthouse.
