Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 1 - Introduction - 1.1 Some Basic Mathematical Models; Direction Fields - Problems - Page 8: 22


$\frac{dv}{dt} = -k V^{2/3}$, where $k$ is some constant.

Work Step by Step

A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation expressing the volume as a function of time. Solution: Volume of sphere: $V=\frac{4}{3}\pi r^{3}$. Surface area of sphere: $A=4\pi r^{2}$ Radius as a function of volume: $r =(\frac{3}{4} \frac{1}{\pi} V)^{1/3}$ Thus, $\frac{dV}{dt}$~$-4\pi[(\frac{3}{4} \frac{1}{\pi} V)^{1/3}]^{2}$. (The minus sign indicates that the volume decreases with time; and, of course, $V$ and $r$ are functions of time.) So, $\frac{dV}{dt}=k_{1}\cdot 3^{2/3}\cdot 4^{1/3}\cdot\pi^{1/3}V^{1/3}$, where $k_{1}$ is the constant of proportionality. Thus, $\frac{dV}{dt} = -k V^{2/3}$, $k$ a constant.
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