A 175-pound force and a 230-pound force are acting on the same point in directions that differ by 42°.
What is the angle that the resultant makes with
the 175-pound force?
A. 22°
B. 24°
C. 32°
D. 18°
let the 175-lb force point in the x direction. Then the 230-lb force acts at an angle θ = 42°.
The two forces form adjacent sides of a parallelogram, with diagonal reaching to
(175+230cos42°,230sin42°) = (346,154)
The angle φ of the diagonal is found using
tan φ = 154/346, so
φ = 24°
To find the angle that the resultant makes with the 175-pound force, we need to use the law of cosines. The law of cosines states that in any triangle with sides a, b, and c and angle C opposite side c, the following equation is true:
c^2 = a^2 + b^2 - 2ab*cos(C)
In this case, the 175-pound force and the 230-pound force are acting on the same point, so they form a triangle. Let's call the angle between the 175-pound force and the resultant angle A. The angle between the 230-pound force and the resultant will be 180° - A, since the sum of the angles in a triangle is 180°.
We are given that the angle between the two forces is 42°. Let's call the length of the resultant force R. Using the law of cosines, we can write the equation:
R^2 = (175^2) + (230^2) - 2(175)(230)*cos(42°)
Now, let's solve for R:
R^2 = 30625 + 52900 - 2(175)(230)*cos(42°)
R^2 = 83525 - 80500*cos(42°)
R^2 ≈ 83525 - 62334.06
R^2 ≈ 21190.94
R ≈ √21190.94
R ≈ 145.54
Now that we know the length of the resultant force, we can find the angle A using the law of sines. The law of sines states that in any triangle with sides a, b, and c and opposite angles A, B, and C respectively, the following equation is true:
a/sin(A) = b/sin(B) = c/sin(C)
In our case, the opposite angle to the 175-pound force is angle A, and the opposite side to angle A is the length of the resultant force, R. We already know the length of R, so we can write the equation:
175/sin(A) = 145.54/sin(42°)
Let's solve for sin(A):
sin(A) = (175 * sin(42°)) / 145.54
sin(A) ≈ 0.5024
A ≈ arcsin(0.5024)
A ≈ 30.02°
Therefore, the angle that the resultant makes with the 175-pound force is approximately 30.02°.
Only option C. 32° is close to 30.02°, so the correct answer is C. 32°.