Question

s grodcon lexl funcion fur a car manufaetuen Lobor. It the tial amcuntr that can be sent 6n nestae metae and, labor is l00 ond units o1 meh and laber Cost $2 and苏3 respectueuv find the Iun Produchen leci fur the manutacturer C Just fAnd x ondl") b) he p xy -42 2xt2xy ty rhure x is the number of units ma

Answer 1

units of labor costs and units of metal costs . Thus total cost is ; this can be at most
100
. So our problem is to

maximize 2x + 2ry+y subject to 3r +2y< 100

We form Lagrangian:

$L(x,y,b)=2x+2xy+y+b(3x+2y-100)$

The partial derivatives are

OL =2x + 1 + 26,

First order conditions and complimentary slackness require

$0=2+2y+3b,~~~~~0=2x+1+2b,~~~~~~0=b(3x+2y-100)$

Solving the first two equations, we get $x=-0.5-b,~~y=-1-1.5b$ . Since can take only non-negative values, we conclude from $x=-0.5-b,~~y=-1-1.5b$ that can not be zero. Hence, from
6(3x + 2y-100) = 0
we conclude that $3x+2y-100=0$. Using $x=-0.5-b,~~y=-1-1.5b$ we get

1.5-3b-2-3b-100 6b-103.5 > -6b103.5

which implies $b=-17.25$. Therefore, from $x=-0.5-b,~~y=-1-1.5b$ we get

$x=16.75,~~~y=24.875$