A tourist looks out from the crown of the Statue of Liberty, approximately 250 ft. above the ground. The tourist sees a ship coming into the harbor and measures the angle of depressions 18°. Find the distance from the base of the statue to the ship to the nearest foot.
Let's use the tangent function to solve this problem. In a right triangle, the tangent of an angle is equal to the ratio of the opposite side (height) to the adjacent side (distance from the base of the statue to the ship). We are given the angle of depression (18°) and the height (250 ft). We want to find the distance (d) from the base of the statue to the ship.
First, we need to find the angle of elevation (α) of the ship from the base of the statue. Since the angle of elevation and the angle of depression are complementary angles, α = 90° - 18° = 72°.
Now we can use the tangent function to find the distance:
tan(α) = height / distance
tan(72°) = 250 / d
To solve for d, we can multiply both sides by d and then divide by tan(72°):
d = 250 / tan(72°)
Use a calculator to find the tangent of 72 degrees:
tan(72°) ≈ 3.07768
Now divide 250 by the tangent:
d ≈ 250 / 3.07768
d ≈ 81.17
So, the distance from the base of the statue to the ship is approximately 81 feet.