To determine the height of the lighthouse given the distance from the boat to the lighthouse and the angle of elevation, we need to use trigonometric functions based on the right triangle formed by the height of the lighthouse (opposite side), the horizontal distance (adjacent side), and the hypotenuse.
Given:
- Distance from the boat to the lighthouse (adjacent side) = 120 meters
- Angle of elevation = [tex]\(40^{\circ}\)[/tex]
- Height of the lighthouse (opposite side) = [tex]\(y\)[/tex]
Since we have the adjacent side and need to find the opposite side, the appropriate trigonometric function to use is the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Plugging in the given values:
[tex]\[
\tan(40^{\circ}) = \frac{y}{120}
\][/tex]
Rearranging this equation to solve for [tex]\(y\)[/tex] (the height of the lighthouse):
[tex]\[
y = 120 \cdot \tan(40^{\circ})
\][/tex]
Hence, the correct equation that represents the situation is:
[tex]\[
\tan 40^{\circ}=\frac{y}{120}
\][/tex]
So, the correct option is:
[tex]\[
\tan 40^{\circ}=\frac{y}{120}
\][/tex]
