   Chapter 14.8, Problem 42E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.42. Exercise 5252. The base of an aquarium with given volume V is made of slate and the sides are made of glass. If slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials.

To determine

To find: The dimensions of the aquarium that minimize the cost of the materials by using Lagrange multipliers.

Explanation

Given:

The base of an aquarium is made of slate and the sides are glass where the slate costs 5 times as much as glass.

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 an aquarium is V=xyz where x>0,y>0,z>0

Therefore, the cost will be, C=5xy+2(xz+yz).

Thus, the minimize function f(x,y,z)=xyz subject to the constraint g(x,y,z)=5xy+2(xz+yz)=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)=λgx(5xy+2(xz+yz)),gy(5xy+2(xz+yz)),gz(5xy+2(xz+yz))yz,xz,xy=λ(5y+2z),(5x+2z),(2x+2y)

Thus, the value of f(x,y,z)=λg(x,y,z) is yz,xz,xy=λ(5y+2z),(5x+2z),(2x+2y).

By the definition, yz,xz,xy=λ(5y+2z),(5x+2z),(2x+2y) can be express as follows

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