Calculus 8th Edition

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

Chapter 2 - Derivatives - 2.8 Related Rates - 2.8 Exercises - Page 185: 11

Answer

\[\frac{dz}{dt}=-18\]

Work Step by Step

$x^2+y^2+z^2=9$ ___(1) Differentiate (1) with respect to $t$ $2x\large\frac{dx}{dt}$+$2y$ $\large\frac{dy}{dt}$+$2z$ $\large\frac{dz}{dt}=0$ ___(2) Using given data $\large\frac{dx}{dt}=5\;,\;\frac{dy}{dt}=4\;,\;$ $\small(x,y,z)=(2,2,1)$ (2) becomes $2(2)(5)+2(2)(4)+2(1)\Large\frac{dz}{dt}$=0 $-36=2\Large\frac{dz}{dt}$ $\Large\frac{dz}{dt}$=$-18$ Hence, $\Large\frac{dz}{dt}$=$-18$.
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