Charles C. answered • 03/20/16

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To minimize labor costs, we want the optimal mix of level I and level II workers. This way we will stay within our desired hours constraint and also produce our desired number of items per week.

Let m be the hours used by level I workers

Let n be the hours used by level II workers.

From the problem, we know

So m + n >= 2600

15m + 22n >= 45000

So n= 2600 - m

Plug in: 15m+ 22(2600-m) > 45000

15m+ 22*2600 - 22m > 45000

We want to solve this equation and get m. Then plug into m + n = 2600 to get n. This will give you the hours needed for each skill level.

Let's continue to solve the above equation

15m -22m > 45000 - 22*2600

-7m > 45000 - 22*2600

-7m > -12200

when we divide by a negative number, we flip the inequality too

So m < -12200 / -7

The negative signs cancel, we get m < 1742.85

So we round down to the nearest whole number, and get m = 1742

We go back to m + n = 2600

1742 + n = 2600

n = 2600 - 1742

n = 858

So m is 1742 hours weekly, and n is 858 hours weekly.

We can also check our work with the equation

15m + 22n >= 45000

The left hand side is 15*1742 + 22*858

26130 + 18876

45006

This is at least 45000, so our answer is correct.