An autonomous underwater vehicle moves along a straight horizontal path with an initial speed \(v_0\). At time \(t = 0\), its propulsion system turns off, and it experiences a resistive acceleration given by \(a = -k v^2\), where \(k\) is a positive constant with units of \(\text{m}^{-1}\) and \(v\) is the instantaneous speed. Which of the following expressions represents the time interval \(\Delta t\) required for the vehicle's speed to decrease from \(v_0\) to \(\dfrac{v_0}{3}\)?