To solve this problem, we need to find the height of the lighthouse given the distance from the boat to the lighthouse and the angle of elevation from the boat to the top of the lighthouse.
Given:
- Distance from the boat to the lighthouse, [tex]\(d = 110\)[/tex] meters
- Angle of elevation, [tex]\(\theta = 44^{\circ}\)[/tex]
- We need to find the height of the lighthouse, [tex]\(h\)[/tex]
Let's review the trigonometric functions and determine which one is appropriate to use in this scenario. The angle of elevation forms a right triangle, with:
- The distance from the boat to the lighthouse as the adjacent side
- The height of the lighthouse as the opposite side
- The line of sight (hypotenuse) going from the boat to the top of the lighthouse
We use the tangent function because:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In this context:
[tex]\[
\tan(44^{\circ}) = \frac{h}{110}
\][/tex]
Thus, matching this equation to the given choices:
1. [tex]\(\cos 44^{\circ} = \frac{y}{110}\)[/tex]
2. [tex]\(\cos 44^{\circ} = \frac{110}{y}\)[/tex]
3. [tex]\(\tan 44^{\circ} = \frac{y}{110}\)[/tex]
4. [tex]\(\tan 44^{\circ} = \frac{110}{y}\)[/tex]
The correct equation is:
[tex]\[
\tan 44^{\circ} = \frac{y}{110}
\][/tex]
Therefore, the correct choice is option 3.
