A flat plate of cross-sectional area \(A\) moves in the \(+x\)-direction at a constant speed \(v_0\) through a cloud of stationary dust particles. At time \(t = 0\), the front face of the plate is located at \(x = 0\). The volume mass density of the dust cloud varies with position according to \(\rho(x) = \rho_0 \left(\dfrac{x}{L}\right)^2\), where \(\rho_0\) and \(L\) are positive constants. As the dust particles collide with the plate, they stick to its surface. Which of the following expressions gives the magnitude of the external force \(F(t)\) required to keep the plate moving at constant speed \(v_0\) as a function of time \(t\)?