To solve this problem, we can use trigonometry and specifically the concept of tangent.
Let's denote the angle that the crane arm forms with the horizontal as θ.
First, draw a diagram to visualize the situation. You should have a right-angled triangle with the crane arm as the hypotenuse, the distance from the base of the crane to the dock as the adjacent side (9 meters), and the length of the crane as the opposite side (6 meters). The cable forms the hypotenuse of a smaller right-angled triangle, with the base of the crane forming the adjacent side (9 meters) and the length of the cable forming the opposite side (15 meters).
Using the tangent function, we have:
tan(θ) = opposite / adjacent
In this case, the opposite side is the length of the crane arm (6 meters) and the adjacent side is the distance from the base of the crane to the dock (9 meters).
Therefore:
tan(θ) = 6 / 9
To find the value of θ, we can take the inverse tangent (arctan) of both sides:
θ = arctan(6/9)
Using a calculator, we can find:
θ ≈ 33.69 degrees
Hence, the acute angle that the crane arm forms with the horizontal is approximately 33.69 degrees.