   Chapter 14.8, Problem 41E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.41. Exercise 5151. Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is a constant c.

To determine

To find: The dimensions of a rectangular box of maximum volume by using Lagrange multipliers.

Explanation

Given:

The sum of the lengths 12 edges of a rectangular box is a constant c.

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 12 edges is equally divided among the dimensions x, y, z and its sum is c.

That is, 4x+4y+4z=c.

Thus, the maximize function f(x,y,z)=xyz subject to the constraint g(x,y,z)=4x+4y+4z=c.

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)<

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