1.) Newjet, Inc. manufactures inkjet printers and laser printers. The company ha

Question

1.) Newjet, Inc. manufactures inkjet printers and laser printers. The company has the capacity to make 70 printers per day, and it has 120 hours of labor per day available. It takes 1 hour to make an inkjet printer and 3 hours to make a laser printer. The profits are $70 per inkjet printer and $90 per laser printer. Find the number of each type of printer that should be made to give maximum profit, and find the maximum profit.

2.) 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 168 per day, and the maximum number of painting hours available is 56 per day. If the profit on the deluxe model is $27 per unit and the profit on the economy model is $21 per unit, how many units of each model will maximize profit?

3.) Solve the linear programming problem. Restrict x 0 and y 0.

Minimize g = 43x + 25y subject to

(x, y)

Explanation / Answer

1.

Let x = the number of inkjet printers to produce per day,
and y = the number of laser printers to produce per day.

Of course, our first two constraints are
x 0, y 0
because we can't make fewer than zero of either. These limit the feasible region to the first quadrant, between the upper y-axis and the right end of the x-axis,
with a vertex (a corner) at the origin: (0,0).

Then we have
x + y 70 [the limit on printers (of either type) made per day]
x + 3y 120 [the limit on hours of labor available]

The first of these sets an upper bound (a "ceiling" limit) on the line
x + y = 70
slanting down from (0,70) to (70,0).

The second sets another upper bound on the line
x + 3y = 120
slanting down from (0,40) to (120,0).

Since we have to obey BOTH of these, the more restrictive one applies,
so we have vertices at (0,40) and (70,0).

These two ceilings cross at (45,25), which is the final vertex of the feasible region.
It's a quadrilateral with vertices
(0,0), (0,40), (45,25), (70,0)

The profit we're supposed to maximize is a function of both x and y:
P(x,y) = 40x + 70y

Because the constraints and the profit are all linear (graphing as straight lines),
we need only check the profits at the vertices to find the maximum:
P(0,0) = 40 [of course!]
P(0,40) = $2800
P(45,25) = $1800 + $1750 = $3550
P(70,0) = $2800

The maximum profit per day is $3550,
obtained by making 45 inkjet printers and 25 laserjet printers per day.

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