Cars on a certain roadway travel on a circular arc of radius $r .$ In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude $\theta$ from the horizontal (see figure). The banking angle must satisfy the equation $r g \tan \theta=v^{2},$ where $v$ is the velocity of the cars and $g=32$ feet per second per second is the acceleration due to gravity. Find the relationship between the related rates $d v / d t$ and $d \theta / d t$ .