need help solving step by step

Question

Acme Business Company has two skill levels of production workers. A level I worker is paid $10.60 an hour and produces 15 items per hour. The level II worker is paid $14.25 per hour and produces 22 items per hour. The company is required to use at least 2600 employee hours per week, and it must produce at least 45,000 items per week. How many hours per week should the company us each skill level to minimize labor costs?

Answer 1

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.