Lagrange Multipliers and Constrained Optimization

 

 

Example:   A rectangular box with an open top is to be designed for a volume of

700 in3.  The material for its bottom cost  7¢ /in2  and the material for its four vertical

sides costs 5¢ /in2.    Find the dimensions of the box that will minimize the cost of

the materials used in constructing the box.

 

 

 

 

 

Strategy:  The first step is to determine the expression for the function and the

expression for the constraint.  In this case the function to be minimized is the cost

function.  Let the box have dimensions  a, b, and c with the dimensions of the bottom

of the box being  a  by  b.   The dimension of the two sides are then   a  by  c    and   

b  by  c.

 

 

 

                       

 

 

Strategy:  Determine the cost function, F, and the constraint function C.

 

   F(a,b,c)  =  7 a b  +  2(5) a c  +  2(5) b c    =   7 a b  +  10 a c  +  10  b c

 

The constraint function is C(a, b, c).  In this example the box is to be designed so

that the volume equals 700  in3 .  So the constraint function is:

 

                            C(a, b, c)  =  a b c  -  700   =  0

 

 

 

 

Click here to continue with this example.

 




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