Flujo

SOLUTIONS MANUAL



CHAPTER 1





Problem 1.1


|Minimize: f(x,y) = xy |[pic] |
|Subject to: (x-8) (y-12) = 300 | |
|Total no. of variables = 2| |
|No. of equality constraints = 1 | |
|No. of degrees of freedom =1 | |
|Independent variable: y| |

Solution:


Eliminate x using the equality constraint


[pic]
[pic]
[pic]
[pic]

[pic]
[pic]
[pic]

Neglecting the physically unrealizable negative value,

[pic]

[pic]

[pic]

Alternative Solution:

|Minimize: area = (w + 8) ( z + 12)|[pic] |
| | |
|St. wz = 300 | |
|| |
|area = wz + 8z + (2w + 96) | |
| | |
|= 300 + 8z 12 [pic]+96| |
|d(area) = 0 = 8 + 3600 [pic] = 0 | |


z2 = 450
z* = + 21.21
w* = 14.14

x* = 8 + 14.14 = 22.14
y* = 12 + 21.21 = 33.21






Problem 1.2


Since thickness is uniform,we just need to minimize the surface area of the inside of the box.

|Minimize: f = b2 + 4bh |[pic] |
|Subject to: b2h = 1000 |b > 0 |
||h > 0 |
|Total no. of variables = 2 | |
|No. of equality constraints = 1 | |
|No. of degrees of freedom = 1| |
|Independent variable = b | |

Solution:

Eliminate h using the equality constraint

[pic]

[pic]

[pic]

[pic]
[pic]minimum.

[pic]

Note: Another viewpoint. Let Δt = thickness ofmaterial. If by material, the volume is used, then the volume of a side is (b) (Δt) (h) and of the bottom is (b) (Δt) (b) so that the objective function would be [pic].



Problem 1.3


|Maximize: A = bh |[pic] |
|Subject to: h = 10-(b/2)2 and [pic] |...