Answer :
Final answer:
Using the tangent function with the angle of depression of 12 degrees and the distance to the boat at 80 meters, the height of the lighthouse can be calculated. After solving, however, the answer does not match the given options, indicating a possible miscalculation or the need to double-check the calculation.
Explanation:
To determine how tall the lighthouse is, based on the angle of depression and the distance to the boat, we can use trigonometry. Given an angle of depression of 12 degrees and a horizontal distance of 80 meters, we consider the lighthouse, the boat, and the line connecting the boat to the lighthouse to form a right-angled triangle. In this case, the height of the lighthouse is the opposite side, and the distance to the boat is the adjacent side of the right triangle.
We can use the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle, to calculate the height:
Tan(12 degrees) = Opposite / AdjacentOpposite = Tan(12 degrees) \( \times \) 80 m
Using a calculator to find Tan(12 degrees) and then multiplying by 80, we get the height of the lighthouse:
Opposite = 0.2125 \( \times \) 80 m = 17 m (approximately)
However, none of the provided choices is 17 m, so we need to double-check the calculation to ensure it's correct.