a boat heading out to sea start out at point a at a horizontal distance of 1035 feet from a lighthouse/the shore from that point the boats crew measures the angle of elevation to the lighthouses beacon-light from that point to be 8 at some later time the crew measures the angle of elevation from point B to be 5 find the distance from point A to point B round your answer to the nearest foot if necessary
To find the distance from point A to point B, we can set up a right triangle with the lighthouse as the vertex of the right angle.
From point A, the angle of elevation to the lighthouse is 8 degrees. This means the angle between the horizontal line from point A to the lighthouse and the line of sight from point A to the top of the lighthouse is 8 degrees.
Using trigonometry, we can calculate the height of the lighthouse above point A:
tan(8 degrees) = height of lighthouse / horizontal distance from point A = height of lighthouse / 1035 feet
height of lighthouse = 1035 feet * tan(8 degrees) ≈ 150.9 feet
Next, from point B, the angle of elevation to the lighthouse is 5 degrees. This means the angle between the horizontal line from point B to the lighthouse and the line of sight from point B to the top of the lighthouse is 5 degrees.
Using trigonometry, we can calculate the distance from point B to the lighthouse:
tan(5 degrees) = height of lighthouse / distance from point B to lighthouse
distance from point B to lighthouse = height of lighthouse / tan(5 degrees) ≈ 1718.89 feet
Therefore, the distance from point A to point B is approximately 1718.89 - 1035 = 683.89 feet, which rounds to 684 feet to the nearest foot.