$6.)(29$ Point$)$ The $20$ kg crate is pulled up the $30$ incline using the cable, winch, and motor $"M$ as shown in the following figure. For a period of time, the tension in the cable is given by $T=(1\mathrm{.}2t^2+140)$ where T $\text{'}$ is in Newtons when $\text{'}T\text{'}$ is in seconds. If the crate has an initial velocity of $v_0=2\frac{m}{s}($up the incline$)$ when $t=0$ and $s=0,$ and the coefficient of friction between the crate and the inclined surface is ${}_k=0\mathrm{.}2,$ determine the distance the crate has moved up the incline after $4$ seconds of travel time.