Final answer:
To calculate the angle of depression from the top of the 250-foot tall lighthouse to the small boat 550 feet away, use trigonometry and the arctan function with opposite side (height of the lighthouse) and adjacent side (distance to the boat). The angle of depression is equal to the angle of elevation due to alternate interior angles.
Step-by-step explanation:
To find the angle of depression from the top of the lighthouse to a small boat, we can use trigonometry. The problem describes a right triangle where the lighthouse is the vertical side, the distance to the boat is the horizontal side, and the angle of depression is the angle at the top of the lighthouse looking down to the boat.
Here is how you calculate the angle of depression:
Define the right triangle: The height of the lighthouse is the opposite side (250 feet), and the distance from the base of the lighthouse to the boat is the adjacent side (550 feet).
Use the tangent function, which relates the opposite side to the adjacent side: tan(angle) = opposite / adjacent.
Calculate the angle: angle = arctan(opposite / adjacent) = arctan(250 feet / 550 feet).
Use a calculator to find the arctan value, which gives you the angle in degrees.
The angle of depression can then be found with a calculator. Remember that the angle of depression from the top of a lighthouse to an object below is equal to the angle of elevation from the object to the top of the lighthouse due to alternate interior angles in parallel lines cut by a transversal (which in this case are the horizontal line through the base of the lighthouse and the line of sight from the top).