Geometrical Meaning of the Derivative

A rope is attached to a pulley mounted on a $15 ft$ tower. The end of the rope is attached to a heavily loaded cart (as shown in figure).

A worker can pull the rope at a rate of $2 ft/s$. The cart is approaching the tower when it is $8 ft$ from the tower at the rate of

At a distance of $4000 ft$ from the launch site, (as shown in the figure) a spectator is observing a rocket being launched. If the rocket lifts off vertically and is rising at a speed of $600 ft/s$ when it is at an altitude of $3000 ft$, the distance between the rocket and the spectator is changing at that instant at the rate: