2018 ONE(d)

A car is being pulled along by a rope attached to the tow-bar at the back of the car.

The rope passes through a pulley, the top of which is 3 m further from the ground than the tow-bar.

The pulley is $x$ m horizontally from the tow-bar, as shown in the diagram above.

The rope is being winched in at a speed of $0.6$ m s${}^{-1}$.

The wheels of the car remain in contact with the ground.

At what speed is the car moving when the length of the rope, $L$, between the tow-bar and the pulley is $5.4$ m?

*You must use calculus and show any derivatives that you need to find when solving this problem.*