Answer:
The distance between the sailboat and the yacht is 655 ft
Step-by-step explanation:
Trigonometric Relations
There are situations where the definitions of the sine, cosine, and tangent of an angle are useful to compute lengths, angles, areas, and other geometric variables. If the situation involves right triangles (with one angle of 90°), there is a hypotenuse and two legs, and this relation holds
[tex]\displaystyle \tan \alpha=\frac{opposite\ leg}{adjacent\ leg}[/tex]
It's also important to remember that if one line crosses another line, then the opposite angles are equal
The situation described in the question can be pictured in the figure below. For ease of construction and calculations, both angles of depression were translated to their equivalent places inside of both triangles. Applying the relation of the tangent, we can say
[tex]\displaystyle \tan 35^o=\frac{200}{AS}[/tex]
Solving for AS
[tex]\displaystyle AS=\frac{200}{\tan 35^o}=\frac{200}{0.7}=286\ ft[/tex]
For the other angle
[tex]\displaystyle \tan 12^o=\frac{200}{AY}[/tex]
Solving for AY
[tex]\displaystyle AY=\frac{200}{\tan 12^o}=\frac{200}{0.2126}=941\ ft[/tex]
Since AY=AS+SY
[tex]SY=941\ ft-286\ ft=655\ ft[/tex]
The distance between the sailboat and the yacht is 655 ft
The distance between the sailboat and the yacht can be determine by using trigonometry relation. In the given situation we will use sine, cosine, tangent formula to compute angle, length and area.
The distance between the sailboat and the yacht is [tex]655\:\rm feet[/tex].
Given:
The cliff height is [tex]200\:\rm ft[/tex].
The angle of depression is [tex]35^{\circ}[/tex].
The below figure shows the given situation.
Consider the triangle [tex]\Delta ASB[/tex], write the formula for [tex]tan[/tex].
[tex]tan35^{\circ} =\frac{AB}{AS}\\tan35^{\circ}=\frac{200}{AS}\\0.7= \frac{200}{AS}\\AS=286 \:\rm ft[/tex]
Consider the triangle [tex]\Delta AYB[/tex], write the formula for [tex]tan[/tex].
[tex]tan12^{\circ} =\frac{AB}{AY}\\tan12^{\circ}=\frac{200}{AY}\\0.2126= \frac{200}{AY}\\AY=941 \:\rm ft[/tex]
From the figure, calculate the value of [tex]SY[/tex].
[tex]AY=AS+SY\\941\:\rm ft=286\:\rm+SY\\SY=655\:\rm ft[/tex]
The distance between the sailboat and the yacht is [tex]SY=655\:\rm feet[/tex].
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