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The Wellbuilt Company produces two types of wood chippers, economy and deluxe. The deluxe model requires 3 hours to assemble and 1/2 hour to paint, and the economy model requires 2 hours to assemble and 1 hour to paint. The maximum number of assembly hours available is 24 per day, and the maximum number of painting hours available is 8 per day. If the profit on the deluxe model is $98 per unit and the profit on the economy model is $72 per unit, how many units of each model will maximize profit? deluxe units economy units?

Respuesta :

Answer:

The maximum profit is reached with 4 deluxe units and 6 economy units.

Step-by-step explanation:

This is a linear programming problem.

We have to optimize a function (maximize profits). This function is given by:

[tex]P=98D+72E[/tex]

being D: number of deluxe units, and E: number of economy units.

The restrictions are:

- Assembly hours: [tex]3D+2E\leq24[/tex]

- Paint hours: [tex]0.5D+1E\leq8[/tex]

Also, both quantities have to be positive:

[tex]D\geq 0\\\\E\geq0[/tex]

We can solve graphically, but we can evaluate the points (D,E) where 2 or more restrictions are saturated (we know that one of this points we will have the maximum profit)

[tex](8;0) \rightarrow  P=98*8+72*0= 784\\\\(0;8) \rightarrow P= 98*0+72*8=576\\\\(4;6) \rightarrow P=98*4+72*6=824[/tex]

The maximum profit is reached with 4 deluxe units and 6 economy units.

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