To determine which of the given equations will find the height of the lighthouse, let's understand the relationship between distance, height, and angle of elevation.
We start with trigonometric principles. If we have:
- The distance from the boat to the lighthouse (let's denote this as [tex]\( d \)[/tex]) is 115 meters.
- The angle of elevation to the top of the lighthouse (denote this as [tex]\( \theta \)[/tex]) is 42 degrees.
- The height of the lighthouse (denote this as [tex]\( h \)[/tex]) is what we need to find.
For such a scenario in right-angled triangle trigonometry, the tangent function ([tex]\(\tan\)[/tex]) relates the angle of elevation ([tex]\(\theta\)[/tex]) to the height ([tex]\( h \)[/tex]) and the distance ([tex]\( d \)[/tex]):
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In this case:
- The "opposite" side is the height of the lighthouse ([tex]\( h \)[/tex]).
- The "adjacent" side is the distance from the boat to the lighthouse ([tex]\( d \)[/tex]) which is 115 meters.
Thus, our equation becomes:
[tex]\[ \tan(42^{\circ}) = \frac{h}{115} \][/tex]
So, out of your given options, the correct equation is:
[tex]\[
\tan 42^{\circ}=\frac{y}{115}
\][/tex]
