Answer:
The total force on the object is approximately 19.52 lbs.
The smallest angle of the triangle is approximately 20.3°.
The largest angle in the triangle is approximately 37.2°.
The remaining angle of the triangle is approximately 122.5°.
Explanation:
To solve this problem, we can use vector addition to find the total force on the object.
We can start by representing the two forces as vectors. Let the vector representing the 12 lbs force be F1 and the vector representing the 5.66 lbs force be F2.
We can then break down each vector into its horizontal and vertical components using trigonometry. Since each force makes a 72.5-degree angle with the horizontal, we have:
F1x = 12 lbs * cos(72.5°) ≈ -3.01 lbs (since the force is pointing to the left)
F1y = 12 lbs * sin(72.5°) ≈ 11.60 lbs (since the force is pointing upward)
F2x = 5.66 lbs * cos(72.5°) ≈ -1.87 lbs (since the force is pointing to the left)
F2y = 5.66 lbs * sin(72.5°) ≈ 7.25 lbs (since the force is pointing upward)
We can then add the horizontal and vertical components of each force separately to get the total force vector:
Fx = F1x + F2x ≈ -4.88 lbs
Fy = F1y + F2y ≈ 18.85 lbs
The total force vector has a magnitude of:
|F| = sqrt(Fx^2 + Fy^2) ≈ 19.52 lbs
So the total force on the object is approximately 19.52 lbs.
To find the angles of the triangle formed by the two forces and the resultant force, we can use the law of cosines:
cos(A) = (B^2 + C^2 - A^2) / (2 * B * C)
where A, B, and C are the lengths of the sides of the triangle opposite to angles A, B, and C, respectively.
Using this formula, we can find:
The smallest angle of the triangle:
cos(A) = (5.66^2 + 19.52^2 - 12^2) / (2 * 5.66 * 19.52) ≈ 0.933
A ≈ 20.3°
The largest angle in the triangle:
cos(C) = (12^2 + 19.52^2 - 5.66^2) / (2 * 12 * 19.52) ≈ 0.785
C ≈ 37.2°
The remaining angle of the triangle:
B = 180° - A - C ≈ 122.5°
Therefore, the answers are:
The total force on the object is approximately 19.52 lbs.
The smallest angle of the triangle is approximately 20.3°.
The largest angle in the triangle is approximately 37.2°.
The remaining angle of the triangle is approximately 122.5°.