Mathematics

Problem

The volume of a solid cylindrical rod is

V = \pi r^{2}L

, where the radius

r

and the length

L

vary with time. At one moment,

r = 0.50\text{ cm}

L = 20\text{ cm}

\dfrac{dr}{dt} = 0.010\text{ cm/min}

, and

\dfrac{dV}{dt} = 0.80\text{ cm}^{3}\!\big/\!\text{min}

. What is

\dfrac{dL}{dt}

at this instant? a)

\displaystyle\frac{0.8}{\pi}-0.2\text{ cm/min}

\displaystyle\frac{3.2}{\pi}-0.8\text{ cm/min}

\displaystyle0.8\pi-3.2\text{ cm/min}

\displaystyle\frac{3.2}{\pi}+0.8\text{ cm/min}

Intuitive explanation

This is a related rates problem where multiple quantities change together through a constraint equation. You're given how the volume formula $V = \pi r^2 L$ links radius and length, and you know the rates of change of $V$ and $r$ —so you need to find the rate of change of $L$ . Differentiate the volume formula with respect to time using the product rule, substitute all the known values at that instant, and solve for the unknown rate $\dfrac{dL}{dt}$ .

Intuitive explanation

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