Two forces of magnitudes 4.0 newtons and 3.0 newtons pull on a box. the forces make an angle of 40° with each other. what is the magnitude of a third force that must be applied to keep the box in equilibrium?

6.6 N

Explanation:

Let's take the direction of the force of 4.0 N as positive x-direction. This means that the force of 3.0 N is at 40 degrees above it. So the components of the two forces along the x- and y-directions are:

So the resultant has components

So the magnitude of the resultant is

And in order for the body to be balanced, the third force must be equal and opposite (in direction) to this force: so, the magnitude of the third force must be 6.6 N.