The volume v of a right circular cylinder of radius r and heigh h is v = pi r^2 h 1. how is dv/dt related to dr/dt if h is constant and r varies with time? 2. how is dv/dt related to dh/dt if r is constant and h varies with time? 3. how is dv/dt related to dh/dt and dr/dt if both h and r vary with time?

In general, the volume



has total derivative



If the cylinder's height is kept constant, then and we have



which is to say, and are directly proportional by a factor equivalent to the lateral surface area of the cylinder ().

Meanwhile, if the cylinder's radius is kept fixed, then



since . In other words, and are directly proportional by a factor of the surface area of the cylinder's circular face ().

Finally, the general case ( and not constant), you can see from the total derivative that is affected by both and in combination.