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?

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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)**

**P=2 ad M=2**

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