   Chapter 14.8, Problem 37E

Chapter
Section
Textbook Problem

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.37. Exercise 4747. Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.

To determine

To find: The maximum volume of a rectangular box that is inscribed in a sphere by using Lagrange multipliers.

Explanation

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 equation of a sphere be x2+y2+z2=r2 where x, y, z are parameters and r is the radius.

Thus, the dimensions are 2x,2yand2z.

Then the volume of the rectangular box is, V=f(x,y,z)=8xyz.

Thus, the maximize function f(x,y,z)=8xyz subject to the constraint g(x,y,z)=x2+y2+z2=r2.

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(8xyz),fy(8xyz),fz(8xyz)=λgx(x2+y2+z2),gy(x2+y2+z2),gz(x2+y2+z2)8yz,8xz,8xy=λ2x,2y,2z

Thus, the value of f(x,y,z)=λg(x,y,z) is 8yz,8xz,8xy=λ2x,2y,2z

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