From the top of a lighthouse 75 feet high, the cosine of the angle of depression of a boat out
at sea is 4. To the nearest foot, how far is the boat from the base of the lighthouse?
The cosine can't be greater than 1.
cosA = 0.4,
A = 66.4 deg.
tan66.4 = 75 / d,
d = 75 / tan66.4 = 33 Ft.
To solve this problem, we need to use trigonometry and the concept of angle of depression.
The angle of depression is the angle between the line of sight from the observer (the top of the lighthouse) to the object (the boat) and a horizontal line. In this case, the lighthouse is 75 feet high, and the cosine of the angle of depression is given as 4.
The cosine of an angle is equal to the adjacent side divided by the hypotenuse in a right triangle. In this case, the adjacent side is the distance from the base of the lighthouse to the boat, and the hypotenuse is the distance from the top of the lighthouse to the boat.
Let's assume the distance from the base of the lighthouse to the boat is x feet. The distance from the top of the lighthouse to the boat is then x + 75 feet.
According to the given information, the cosine of the angle of depression is 4, which means:
cos(angle) = adjacent / hypotenuse
4 = x / (x + 75)
To solve for x, we can cross multiply:
4(x + 75) = x
4x + 300 = x
3x = -300
x = -100
Since distance cannot be negative, we discard the negative solution. Therefore, the boat is approximately 100 feet from the base of the lighthouse.