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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?

User Andrean
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1 Answer

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In general, the volume


V=\pi r^2h

has total derivative


(\mathrm dV)/(\mathrm dt)=\pi\left(2rh(\mathrm dr)/(\mathrm dt)+r^2(\mathrm dh)/(\mathrm dt)\right)

If the cylinder's height is kept constant, then
(\mathrm dh)/(\mathrm dt)=0 and we have


(\mathrm dV)/(\mathrm dt)=2\pi rh(\mathrm dt)/(\mathrm dt)

which is to say,
(\mathrm dV)/(\mathrm dt) and
(\mathrm dr)/(\mathrm dt) are directly proportional by a factor equivalent to the lateral surface area of the cylinder (
2\pi r h).

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


(\mathrm dV)/(\mathrm dt)=\pi r^2(\mathrm dh)/(\mathrm dt)

since
(\mathrm dr)/(\mathrm dt)=0. In other words,
(\mathrm dV)/(\mathrm dt) and
(\mathrm dh)/(\mathrm dt) are directly proportional by a factor of the surface area of the cylinder's circular face (
\pi r^2).

Finally, the general case (
r and
h not constant), you can see from the total derivative that
(\mathrm dV)/(\mathrm dt) is affected by both
(\mathrm dh)/(\mathrm dt) and
(\mathrm dr)/(\mathrm dt) in combination.
User Chen Wang
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