# Chapter 3 - Differentiation - 3.9 Related Rates - Exercises - Page 159: 14

$0.533\pi\, m^{3}/min$

#### Work Step by Step

$r=2\,m, h=3\,m$$\implies\frac{r}{h}=\frac{2}{3}\implies r=\frac{2}{3}h V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi (\frac{2}{3}h)^{2}h=\frac{4\pi}{27}h^{3} \frac{dV}{dt}=\frac{dV}{dh}\times\frac{dh}{dt}$$=\frac{4\pi}{27}\times3h^{2}\times\frac{dh}{dt}=\frac{12\pi}{27}\times h^{2}\times\frac{dh}{dt}$ When $\frac{dh}{dt}= 0.3\,m/min$ and $h=2\,m$, $\frac{dV}{dt}= \frac{12\pi}{27}\times(2\,m)^{2}\times0.3\,m/min$ $\approx0.533\pi\,m^{3}/min$

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