A helicopter hovers 650 feet above a small island. The figure shows that the angle of depression from the helicopter to point P is 31 degrees How far off the coast, to the nearest foot, is the island? Shown in the image as distance d.
Since we know the angle of depression from the helicopter to point P is 31 degrees, we can conclude that angle P is also 31 degrees (as it is complementary to the angle of depression).
Now, we can use trigonometry to find the length of side d.
Let's denote the height of the helicopter as h and the length of side d as x.
Since the tangent of an angle is equal to the opposite side divided by the adjacent side, we can write:
tan(31 degrees) = h / x
Rearranging the equation, we have:
x = h / tan(31 degrees)
Now, substituting the given values:
x = 650 feet / tan(31 degrees)
Using a calculator:
x ≈ 1167.20 feet
Therefore, the island is approximately 1167 feet off the coast.