Answer:
524 feet.
Step-by-step explanation:
Given: The height of cliff is 200 feet.
Angle of depression at sailboat is 42°
Angle of depression at a yacht is 15°
We will use tangent rule to find distance of sailboat and yacht from cliff´s bottom.
We know, the opposite leg has a height 200 feet.
Formula, [tex]tan \theta = \frac{Opposite\ leg}{adjacent\ leg}[/tex]
First finding distance of sailboat from bottom of cliff.
[tex]tan 42= \frac{200}{adjacent\ leg}[/tex]
⇒ [tex]0.90= \frac{200}{adjacent\ leg}[/tex]
Cross multiplying
we get, Adjacent leg= [tex]\frac{200}{0.90} = 222.22\ feet[/tex]
∴ Distance of sailboat from cliff´s bottom is 87.33 feet.
Now, finding the distance of yatch from bottom of cliff.
[tex]tan 15= \frac{200}{adjacent\ leg}[/tex]
⇒ [tex]0.2679=\frac{200}{adjacent\ leg}[/tex]
Cross multiplying
we get, Adjacent leg= [tex]\frac{200}{0.2679}= 746.5\ feet[/tex]
∴ Distance of yatch from cliff´s bottom is 746.5 feet.
Next, finding difference between sailboat and yatch.
Difference= [tex]Yacht\ distance - sailboat\ distance[/tex]
⇒ Difference= [tex]746.5 \ feet - 222.22\ feet= 524.32 \approx 524\ feet.[/tex]
Hence, the distance between sailboat and yacht is 524 feet.
Distance between the sailboat and the yacht is 524 feet.
From the figure attached,
Let the distance between sail boat and the yacht (CD) = x feet
Applying tangent rule in ΔABD for angle measuring 42°,
tan(42°) = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
= [tex]\frac{AB}{BD}[/tex]
tan(42°) = [tex]\frac{200}{BD}[/tex]
BD = [tex]\frac{200}{\text{tan}(42^\circ)}[/tex]
BD = 222.12 feet
Apply tangent rule in ΔABC for the angle measuring 25°.
tan(15°) = [tex]\frac{AB}{BC}[/tex]
BC = [tex]\frac{200}{\text{tan}(15^\circ)}[/tex]
BC = 746.41 feet
Since, BC = BD + CD
746.41 = 222.12 + x
x = 746.41 - 222.12
x = 524.29 feet
x ≈ 524 feet
Therefore, distance between the sailboat and the yacht is 524 feet.
Learn more,
https://brainly.com/question/14112579
