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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

If the length of the diagonal of a rectangular box must be L, what is the largest possible volume?

To determine

To find: The largest volume of the rectangular box.

Explanation

Given:

The length of the diagonal of the rectangular box is L .

Calculation:

Let the dimensions be x,yandz .

Then, the volume of the rectangular box is V(x,y,z)=xyz where x>0,y>0,z>0 .

The length of the diagonal of the rectangular box is L=x2+y2+z2 .

Thus, the maximum value of the function V(x,y,z)=xyz subject to the equation L=x2+y2+z2 .

The value of z is calculated from the equation L=x2+y2+z2 .

L2=x2+y2+z2z2=L2x2y2z=L2x2y2

Substitute z=L2x2y2 in the function V=xyz .

V=xy(L2x2y2)

Take the partial derivative of the function V with respect x and obtain Vx .

Vx=x(xy(L2x2y2))=yx(xL2x2y2)=y[L2x2y2(1)+x(12)(L2x2y2)12(2x)]=yL2x2y2x2yL2x2y2

Hence, Vx=yL2x2y2x2yL2x2y2 . (1)

Take the partial derivative of the function V with respect y and obtain Vy .

Vy=y(xy(L2x2y2))=xy(yL2x2y2)=x[L2x2y2(1)+y(12)(L2x2y2)12(2y)]=xL2x2y2xy2L2x2y2

Thus, Vy=xL2x2y2xy2L2x2y2 . (2)

Set the above partial derivatives to 0 and find the critical points.

From the equation (1),

yL2x2y2x2yL2x2y2=0y(L2x2y2)x2yL2x2y2=0y(L2x2y2)x2y=0y(L2x2y2x2)=0

Simplify further as follows

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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