The force F (in Newtons) required to move a box of mass m kg in motion by pulling on an attached rope is F(x) = f m g/(cos(x))+ f (sin(x))

where x is the angle between the rope and the horizontal, f is the coefficient of static friction and g = 9.8 m/s^2. Find the angle x that minimizes the required force F, assuming f = .4

To find the angle x that minimizes the required force F, we need to find the value of x that minimizes the function F(x).

1. Start with the given function for force F(x):
F(x) = f * m * g / (cos(x)) + f * (sin(x))

2. Substitute the given value for the coefficient of static friction (f = 0.4) and the acceleration due to gravity (g = 9.8 m/s^2):
F(x) = 0.4 * m * 9.8 / (cos(x)) + 0.4 * (sin(x))

3. To minimize the force F(x), we need to find the critical point(s) by taking the derivative of F(x) with respect to x and setting it equal to zero:
F'(x) = -0.4 * m * 9.8 * sin(x) / (cos(x))^2 + 0.4 * cos(x) = 0

4. Simplify the equation by multiplying both sides by (cos(x))^2:
-0.4 * m * 9.8 * sin(x) + 0.4 * cos(x) * (cos(x))^2 = 0

5. Rearrange the equation:
-0.4 * m * 9.8 * sin(x) = 0.4 * cos(x) * (cos(x))^2

6. Divide both sides by 0.4:
-m * 9.8 * sin(x) = cos(x) * (cos(x))^2

7. Divide both sides by cos(x) (assuming cos(x) is not equal to zero):
-m * 9.8 * tan(x) = (cos(x))^2

8. Rearrange the equation:
(cos(x))^2 + m * 9.8 * tan(x) = 0

At this point, we have a nonlinear equation that cannot be solved analytically. To find the angle x that minimizes the force F, we will need to solve this equation numerically using a numerical method or software.