12. A box of mass m is being pulled by a rope as shown. The force F in the rope as a function of x can be calculated from the equations:

where N and are the normal force and friction coefficient between the box and surface, respectively. Consider the case where kg, m,

, and m/s2.

(a) Use MATLAB to derive an expression for F, in terms of x, h, m, g, and

.

(b) Use the subs command to substitute kg, m, , and m/s2 into the expressions that were derived in part (a).

This will give the force as a function of the distance x.

(c) Use the ezplot command to plot the force F as a function of x, for x starting at 5 and ending at 30 m.

(d) Determine the distance x where the force that is required to pull the box is the smallest, and determine the magnitude of that force.