Please Help!!!!!!

Question

Your company has a small manufacturing plant that makes three types of bottles for selling your salsa: 8 oz, 16 oz, and 32 oz bottles. Each bottle requires the service of three departments: filling, labeling and packaging. For an 8 oz bottle it takes 0.6 hours to fill, 0.6 hours to label it, and 0.2 hours to package it. Then for the 16 oz bottle it takes 1 hour to fill 0.9 hour to label and 0.3 hour to package it. Then for the 32 oz bottle it takes 1.5 hour to fill it 1.2 hours to label it and 0.5 hour to package it. If the filling and the labeling and packaging departments are allowed to use a total of 380,330,and 120 person-hours per week, respectively, determine how many of each type of bottle your company must produce each week for the plant to operate at full capacity.

Answer 1

x = number of 8 oz. bottles

y = number of 16 oz. bottles

z = number of 32 oz. bottles

 

We have to set up a system of equations with the number of hours it takes for each task times the specific bottle type.

 

Filling

0.6x + 1y +1.5z =380

 

Labeling

0.6x + 0.9y + 1.2z = 330

 

Packaging

0.2x +0.3y + 0.5z = 120

 

We must reduce this to 2 equations by eliminating one variable (in this case I chose to eliminate x). 

 

Multiply Labeling by -1 to get -0.6x -0.9y - 1.2z = -330 and add to equation 1 (filling) to get:

 

0.1y +0.3z = 50

 

Multiple Packaging by -3 to get -0.6x -0.9y -1.5z = -360 and again add to equation 1 (filling) to get:

 

0.1y= 20

 

Now we have a system of question of 2 equations instead of three:

 

0.1y + 0.3z = 50

0.1y =20

 

Solve for y = 20/.1= 200. Now solve for z:

 

0.1 (200) +0.3z = 50

0.3z = 30

z= 30/0.3= 100

 

Now plug back into the Filling Equation to solve for x:

 

0.6x + 1y +1.5z = 380

0.6x + 1 (200) + 1.5(100) = 380

0.6x = 30

x = 30/0.6 = 50

 

Therefore, the company should produce 50 8 oz bottles (x), 200 16 oz. bottles (y), and 100 32 oz. bottles to be at full capacity.

 

To check these numbers, you can plug back into any of the original equations (Labelling for example)

 

0.6x + 0.9y + 1.2z = 330

0.6 (50) + 0.9 (200) + 1.2 (100) = 330

330=330 so it checks out.