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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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

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Chapter 14 Solutions

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