# An entrepreneur is having a design group produce at least nine samples of a new kind of fastener that he wants to market.  It costs \$3.00 to produce each metal fastener and \$9.00 to produce each...

An entrepreneur is having a design group produce at least nine samples of a new kind of fastener that he wants to market.  It costs \$3.00 to produce each metal fastener and \$9.00 to produce each plastic fastener.  He wants to have at least two of each version of the fastener and needs to have all the samples 42 hours from now.  It takes 5 hours to produce each metal sample and 1 hour to produce each plastic sample.  To minimize the cost of the samples, how many of each kind should the entrepreneur order? What will be the cost of the samples?

aruv | High School Teacher | (Level 2) Valedictorian

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Mathematical model of  proble is

Let C be the cost. P be the  no. of the plastic fastener and M be the no of metal fastener.

Thus

`C=3M+9P`

s.t.

At least constraints.

`M>=2`

`P>=2`

Time constraints

`5M+P<=42`

So Problem is

Min `C=3M+9P`

s.t.

`M>=2`

`P>=2`

`5M+P<=42`

Now draw the graph  M- along x-axis and P along y-axis

The solution of set of inequality is

P(2,32) ,Q(8,2) and R(2,2)

In graph Q is point of intersection of red and green line.

P is point of intersection of redline and x=2.

R is point of intersection of lines x=2 and y=2

Thus

C at P=3x2+9x32

=294

C at Q=3 x 8+9 x2

=24+18

=42

C at R= 3x2+9x2

=6+18

=24

Thus Min( 294,42,24)=24

Min cost is=\$24  at R(2,2)