   Chapter 14.8, Problem 40E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.40. Exercise 5050. Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.

To determine

To find: The dimensions of the rectangular box with largest volume by using Lagrange multipliers.

Explanation

Given:

The total surface area is 64cm2.

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 surface area of the rectangular box is g(x,y,z)=xy+yz+zx=32.

Thus, the maximize function f(x,y,z)=xyz subject to the constraint g(x,y,z)=xy+yz+zx=32.

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(xy+yz+zx),gy(xy+yz+zx),gz(xy+yz+zx)yz,xz,xy=λ(y+z),(x+z),(x+y)

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

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