   Chapter 14, Problem 64RE

Chapter
Section
Textbook Problem

A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume that can be mailed.

To determine

To find: The dimensions of the package with largest volume.

Explanation

Given:

The given volume is V=xyz where x, y, z are sides of the box.

The perimeter is x+2y+2z108 where x>0,y>0,z>0 and the lengths of y ,z is 2y, 2z.

Calculation:

The given function is f(x,y,z)=xyz .

The perimeter is maximized into g(x,y,z)=x+2y+2z108 .

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(x+2y+2z108),gy(x+2y+2z108),gz(x+2y+2z108)yz,xz,xy=λ1,2,2

Thus, the value of f(x,y,z)=λg(x,y,z) is yz,xz,xy=λ1,2,2

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