Answer: Then the magnitude of the force is 37.86N, and the direction is 54.35°
Step-by-step explanation:
We can write the forces as vectors.
We know that Annie pushes with a magnitude of 80N in a direction of 133° (Remember that the angles are always measured from the x-axis)
The components of this force, (Ax, Ay), are then:
Ax = 80N*cos(133°)
Ay = 80N*sin(133°)
And we know that Josie pushes with a magnitude of 95N in direction of 290°
The components of this force, (Jx, Jy), are:
Jx = 95N*cos(290°)
Jy = 95N*sin(290°)
When we add these forces, the total force acting on the ball is:
F = (80N*cos(133°) , 80N*sin(133°)) + (95N*cos(290°), 95N*sin(290°))
F = (80N*cos(133°) + 95N*cos(290°), 80N*sin(133°) + 95N*sin(290°))
Now, the third kitten wants to do a force K, in a direction θ, such that the net force acting on the ball is zero.
Then we must have that, each component of the force of the third cat (K*cos(θ) on the x-axis and k*sin(θ) on the y-axis), is such that:
K*cos(θ) + 80N*cos(133°) + 95N*cos(290°) = 0
k*sin(θ) + 80N*sin(133°) + 95N*sin(290°) = 0
Now we need to solve that system for k and θ
if we simplify the equations we get:
k*cos(θ) - 22.07N = 0
k*sin(θ) -30.76N = 0
Now we can rewrite them as:
k*cos(θ) = 22.07N
k*sin(θ) = 30.76N
Now we can take the quotiet between both equations to get:
(k*sin(θ))/(k*cos(θ)) = 30.76N/22.07N
Tan(θ) = 1.394
θ = Atan(1.394) = 54.35°
Now that we know the angle, we can find the value of the magnitude k, by using one of the two equations of the system:
k*cos(54.35°) = 22.07N
k = 22.07N/cos(54.35°) = 37.86N
Then the magnitude of this force is 37.86N, and the direction is 54.35°
