Final answer:
Using trigonometry and the tangent of the angle of elevation, we can find the distance between the boat and the lighthouse. By the formula distance = height / tan(angle of elevation), the distance is approximately 134 meters, making option d correct.
Step-by-step explanation:
To find out how far the boat is from the lighthouse, we can use trigonometry. The height of the lighthouse is given as 46 meters, and the angle of elevation from the boat to the top of the lighthouse is 19 degrees. We are looking for the distance between the boat and the base of the lighthouse, which forms the adjacent side of a right-angled triangle, where the lighthouse's height is the opposite side.
The tangent of an angle in a right-angled triangle is the ratio between the opposite side and the adjacent side. In mathematical terms, tangent (angle of elevation) = opposite side / adjacent side. Using the given angle of 19 degrees, we have: tan(19°) = 46 / distance to the lighthouse. By rearranging this equation to solve for the distance, we get the distance to the lighthouse as the height divided by the tangent of the angle.
Therefore, the distance to the lighthouse is given by:
distance = 46 / tan(19°). When calculated, this gives a distance of approximately 134 meters, which makes option d (134 m) the correct answer.