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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.

43. Exercise 55

55. 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 by using Lagrange multipliers.

Explanation

Given:

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

Definition used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z). This equation can be expressed as fx=λgx, fy=λgy, fz=λgz and g(x,y,z)=k”.

Calculation:

Let the dimensions be x,yandz.

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

The length of the diagonal of the rectangular box is g(x,y,z)=L=x2+y2+z2.

Thus, the maximum value of the function f(x,y,z)=xyz subject to the constraint g(x,y,z)=x2+y2+z2=L2.

The Lagrange multipliers f(x,y,z)=λg(x,y,z) is computed as follows,

f(x,y,z)=λg(x,y,z)fx,fy,fz=λgx,gy,gzfx(xyz),fy(xyz),fz(xyz)=λgx(x2+y2+z2),gy(x2+y2+z2),gz(x2+y2+z2)yz,xz,xy=λ2x,2y,2z

Thus, the value of f(x,y,z)=λg(x,y,z) is yz,xz,xy=λ2x,2y,2z

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Chapter 14 Solutions

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