Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.9 Related Rates - Exercises - Page 159: 15



Work Step by Step

The volume of a circular cone is $V=\frac{1}{3}\pi r^{2}h$ $\frac{dV}{dt}=\frac{1}{3}\pi \times\frac{d}{dt}(r^{2}h)$ Applying the product rule, we have $\frac{d}{dt}(r^{2}h)=2r\times\frac{dr}{dt}\times h+r^{2}\times\frac{dh}{dt}$ Given: $r=10\,cm$, $h=20\,cm$, $\frac{dr}{dt}=2\,cm/s$ and $\frac{dh}{dt}=2\,cm/s$ Substituting the given values, we get $\frac{d}{dt}(r^{2}h)=1000\,cm^{3}/s$ Then, $\frac{dV}{dt}=\frac{1}{3}\pi\times1000\,cm^{3}/s=\frac{1000\pi}{3}cm^{3}/s$
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