Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.8 Lagrange Multipliers - 14.8 Exercises - Page 1018: 42


Dimensions of a box are: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$

Work Step by Step

Need to apply Lagrange Multipliers Method to determine the dimensions of a rectangular box of maximum volume. we have $\nabla f=\lambda \nabla g$ The volume of a box is $f=V=xyz$ and From the given question let us consider $\nabla f=\lt 10x+4z,4x \gt$ and $\lambda \nabla g=\lambda \lt 2xz, x^2 \gt$ Using the constraint condition we get, $10x+4z=\lambda 2xz$ $4x=\lambda x^2;$ $V=x^2z$ Simplify to get the value of $z$. we get $z=\dfrac{5x}{2}$ and $V=x^2z$ or, $x=\sqrt[3] {\dfrac{2}{5}}V$ Dimensions of a box are: $x=\sqrt[3] {\dfrac{2}{5}}V$,$y=\sqrt[3] {\dfrac{2}{5}}V$ and $z=\dfrac{5}{2}\sqrt[3] {\dfrac{2}{5}}V$
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