$\"\"$

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Let $\"\"$ be the number of the metal fastener.

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Let $\"\"$ be the number of the plastic fastener.

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Cost of each metal fastener is $\"\"$ .

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Cost of each metal fastener is $\"\"$ .

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Total cost of six samples is $\"\"$ .

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Hence the constraint is $\"\"$ .

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Entrepreneur wants to produce at least two of each samples.

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Hence the constraints are $\"\"$ .

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It takes $\"\"$ hours to produce metal fastener and $\"\"$ hours to produce plastic fastener

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It does not exceed $\"\"$ hours.

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Hence the constraint is $\"\"$ .

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$\"\"$

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The objective function is $\"\"$ .

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The constraints are

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$\"\"$

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Graph :

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Graph the inequalities and shade the required region.

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$\"\"$

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Note : The shaded region is the set of solution points for the objective function.

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Observe the graph:

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Tabulate the solutions of each of two system of inequalities and obtain the intersection points.

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System of boundary

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equations

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$\"\"$

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$\"\"$

\

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Solution (vertex points)

\"\"

\"\"

\"\"

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$\"\"$

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$\"\"$

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Find the value of objective function at the solution points.

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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Observe the values of $\"\"$ :

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The minimum value of $\"\"$ is $\"\"$ at $\"\"$ .

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Therefore, entrepreneur should make $\"\"$ metal fastener and $\"\"$ plastic fastener for total cost of $\"\"$ .

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$\"\"$

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Entrepreneur should make $\"\"$ metal fastener and $\"\"$ plastic fastener for total cost of $\"\"$ .