   Chapter 14.8, Problem 44E

Chapter
Section
Textbook Problem

Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm2 and whose total edge length is 200 cm.

To determine

To find: The maximum and minimum volumes of a rectangular box.

Explanation

Given:

The total surface area of a rectangular box is 1500cm2.

The total edge length of a rectangular box is 200cm.

Definition used:

“The Lagrange multipliers defined as f(x0,y0,z0)=λg(x0,y0,z0)+μh(x0,y0,z0). This equation can be expressed as fx=λgx+μhx, fy=λgy+μhy, fz=λgz+μhz and g(x,y,z)=k, h(x,y,z)=c”.

Calculation:

Let the dimensions of the rectangular box 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)=2xy+2yz+2zx=1500.

The total edge length of the rectangular box is h(x,y,z)=4x+4y+4z=200.

Thus, the maximize function f(x,y,z)=xyz is subject to the constraints g(x,y,z)=xy+yz+zx=750 and h(x,y,z)=x+y+z=50.

The Lagrange multipliers f(x,y,z)=λg(x,y,z)+μh(x,y,z) is computed as follows.

f(x,y,z)=λg(x,y,z)+μh(x,y,z)fx,fy,fz=λgx,gy,gz+μhx,hy,hzfx(xyz),fy(xyz),fz(xyz)=λgx(xy+yz+zx),gy(xy+yz+zx),gz(xy+yz+zx)+μhx(x+y+z),hy(x+y+z),hz(x+y+z)yz,xz,xy=λy+z,x+z,x+y+μ1,1,1

Thus, the value of f(x,y,z)=λg(x,y,z)+μh(x,y,z) is yz,xz,xy=λy+z,x+z,x+y+μ1,1,1.

The result, yz,xz,xy=λy+z,x+z,x+y+μ1,1,1 can be expressed as follows.

yz=λ(y+z)+μ (1)

xz=λ(x+z)+μ (2)

xy=λ(x+y)+μ (3)

Subtract the equation (2) from (1),

yzxz=[λ(y+z)]+μ[λ(x+z)]μz(yx)=λ(y+zxz)z(yx)=λ(yx)

Since, z(yx)=λ(yx), there are two possibilities such as either z=λorx=y

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Define the derivative f'(a). Discuss two way of interpreting this number.

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 2340, find the indicated limit. 39. limx1x2+82x+4

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Sometimes, Always, or Never: If c is a critical number, then f′(c) = 0.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

For and , a × b =

Study Guide for Stewart's Multivariable Calculus, 8th 