The maximum force that can be applied to the box before it will begin to accelerate is the same as the maximum static friction that the interaction between the box and the surface can generate.
The maximum static friction is given by:
[tex]f=\mu N[/tex]Where μ is the coefficient of friction and N is the magnitude of the normal force between the box and the surface.
The magnitude of the normal force that acts on an object at rest on a horizontal surface is equal to its weight:
[tex]N=mg[/tex]Then, the maximum static friction for this situation is:
[tex]f=\mu mg[/tex]Substitute μ = 0.45, m = 30.0kg and g = 9.81 m/s^2:
[tex]f=(0.45)(30.0\operatorname{kg})(9.81\frac{m}{s^2})=132.4N[/tex]Therefore, the maximum amount of force that can be applied to the box parallel to the surface before the box will begin to accelerate, is:
[tex]132.4N[/tex]