a crane at the edge of a dock is supported by a cable 15 meters long, attached to the dock 9 meters long from the base of the crane. if the crane is 6 meters long, what is the acute angle that the crane arm forms with the horizontal?
As I interpret the data, it appears that the law of cosines will provide our answer. We want the supplement to θ, where
15^2 = 6^2+9^2 - 2(6)(9)cosθ
cosθ = -1
Nope, that doesn't work. Maybe you can just describe the figure, giving labels to the points of interest.
180
To find the acute angle that the crane arm forms with the horizontal, we can use the cosine formula.
Let's label the angle formed by the crane arm and the horizontal as θ.
In a right triangle, the adjacent side is the side next to the angle, and the hypotenuse is the longest side.
In this case, the adjacent side is 9 meters (the distance from the base of the crane to the dock), and the hypotenuse is 15 meters (the length of the cable).
Using the cosine formula: cos(θ) = adjacent / hypotenuse
cos(θ) = 9 / 15
Now we can find the value of cos(θ) by dividing:
cos(θ) ≈ 0.6
To find the angle θ, we can take the inverse cosine (arccos) of cos(θ):
θ = arccos(0.6)
Using a calculator or trigonometric table, we find that θ is approximately 53.13 degrees.
Therefore, the acute angle that the crane arm forms with the horizontal is approximately 53.13 degrees.
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.