Final answer:
The horizontal distance from a ship to the base of a 200-foot-tall lighthouse with an angle of depression of 14° is approximately 802 feet, which is calculated using the tangent function related to the angle of elevation and sides of a right triangle. option D) 802 feet is the correct answer after rounding to the nearest foot is the correct answer.
Step-by-step explanation:
To find the horizontal distance from a ship to the base of a 200-foot-tall lighthouse with an angle of depression of 14°, we can utilize trigonometric principles specifically related to right-angle triangles.
The angle of depression is equal to the angle of elevation from the ship to the top of the lighthouse because alternate interior angles are congruent when two lines are parallel, which in this case is assumed by the level surface of the sea and the hypothetical line from the ship to the base of the lighthouse.
The angle of elevation is thus also 14°. In this model, we can use the tangent function, which relates the angle of a right triangle to the ratio of the opposite side over the adjacent side.
Let's denote the distance from the ship to the lighthouse as x. Using the formula τ = opposite/adjacent, which is tangent(14°) = 200/x. Solving for x gives x = 200/tangent(14°). After calculating, we find that x is approximately 802 feet, which means option D) 802 feet is the correct answer after rounding to the nearest foot.