Question

Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you mame a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. How many of what type of printer should you order if you want to minimize your cost?

Answer 1

Let x represent the number of A printers
Let y represent the number of B printers

Minimize cost = 86x + 130y
subject to
Total printers equn: x + y ≥ 100
Total profit equn: 45x + 35y ≥ 3850
x ≥ 0, y ≥ 0
x and y must be whole numbers.

The vertices of the feasible region are: (0, 100), (100, 0) and (35, 65)

If x = 35 and y = 65 the cost is 11460 and profit is 3850
if x = 100 and y = 0 the cost is 8600 and profit is 4500
If x = 0 and y = 100 the cost is 13000 and profit is 3500

The best result is x = 100 and y = 0

Answer 2

100 printers of Type A

Explanation:

Let x be the no. of printers of type A

Let y be the no. of printers of type B

You expect to sell at least 100 laser printers this month

Equation becomes :
`x+y\geq100` ---1

Cost of 1 printer of Type A = $86

Cost of x printer of Type A = $86x

Cost of 1 printer of type B =$135

Cost of y printer of type B =$135 y

Minimize cost function:
`86x+130y`

Now Profit on 1 Type A printer = $45

Profit on x Type A printer = $45x

Profit on 1 Type B printer = $35

Profit on y Type B printer = $35y

We are given that you need to make at least $3850 profit on them.

So, equation becomes :
`45x+35\geq 3850` ---2

Conditions :
`x\geq 0` ---3 and
`y\geq 0` ---4

Now plotting the lines 1,2,3,4 on the graph

Refer the attached figure

Feasible points are (100,0);(0,100)and(35,65)

Now check which feasible point provides minimum cost.

`86x+130y`

At point (100,0)

`86(100)+130(0)`

`8600`

So, At point (100,0) total cost is $8600.

At point (0,100)

`86(0)+130(100)`

`13000`

So, At point (0,100) total cost is $13000

At point (35,65)

`86(35)+130(65)`

`11460`

So, At point (35,65) total cost is $11460

So, at (100,0) we are getting the minimum cost i.e. $8600.

So, we need to order 100 printers of type A and 0 printers of type B to minimize the cost.