A local family-owned plastic cup manufacturer wants to optimize their production mix in order to maximize their profit. They produce 2 sizes of plastic cups for a chain of stores selling smoothies; 16 oz and 20oz cups. Currently, the smoothie company purchases three times as many 16 oz cups to 20 oz cups.

The profit on a case of 20 oz plastic cups is $20 while the profit on a case of 16 oz plastic cups is $25. 

The cups are manufactured with a machine called a plastic extruder which feeds on plastic resins. Each case of 20 oz cups requires 18 lbs. of plastic resins to produce while 16 oz plastic cups require 14 lbs. per case. The daily supply of plastic resins is limited to at most 1800 pounds.

About 15 cases of either product can be produced per hour. At the moment, the family wants to limit their workday to 8 hours.

A system of linear equations that model this situation:

Let x =                                                                                                

Let y =                                                                                                

An equation to represent the number of cases that can be made in 8 hours:                                                            

An equation to represent the maximum Amount of plastic resin used per day:                                                                      

An equation to represent the number of 16 oz vs 20 oz cups made:                                                                                           

Use either substitution or elimination to solve the system of equation:

What is the solution to this system? (ordered pair)

What does this solution represent?

Graph the system of equations

Find an equation to represent the Profit:  P =            

What is the maximum profit that can be made given the parameters of this system:  

Currently, the smoothie company purchases three times as many 16 oz cups to 20 oz cups, however, they said in the near future they would like to order four times as many 16 oz cups to 20 oz cups.

How many of each type of cup should be made if the smoothie company changes their order? 

What would the profit be for this new situation?

Answer:

To make a system of linear equations that model this situation:

What are the variable x & y ?

Let x = cases of 16 oz cups produced per day

Let y = cases of 20 oz cups produced per day

Use an equation to represent the number of cases that can be made in 8 hours:

Number of cases that can be made in 8 hours:

?x+y=15∗8=120?

Use an equation to represent the maximum Amount of plastic resin used per day:

Maximum Amount of plastic resin used per day:

?14x+18y≤1800?

Use an equation to represent the number of 16 oz vs 20 oz cups made:

Number of 16 oz vs 20 oz cups made: ?x:y=3:1?

So, ?x=3y?

Use either substitution or elimination to solve the system of equation:

we will solve the system by substitution method

As we have ?x:y=3:1?

So, ?x=3y?

?x+y=120? ..... (from (1))

we will substitute value of x, as ?x=3y?

?3y+y=120?

4y=120

?y=4120?? = 30

y = 30

putting this value of y in equation ?x=3y?

?x=3y?

?x? = 3 * 30 =90

x = 90

What is the solution to this system?

• Each case of 20 oz cups requires 18 lbs. of plastic resins to produce while 16 oz plastic cups require 14 lbs. per case
• The daily supply of plastic resins is limited to at most 1800 pounds.

?14x+18y≤1800?

now we will put x = 90 , y = 30 , we will get

?14x+18y=14×90+18×30=1800≤1800?

So the solution is ?(x,y)=(90,30)?

What does this solution represent?

The solution indicates that for having the maximum production level,

The company must have the total numbers of 16 oz of cups and 20 oz of cups to be produced every day will be 90 and 30 units respectively.

Graph : (three lines showing three constraints )

From graph, three lines are intersects at (90,30)

Therefore, solution is x= 90 and y = 30

Find an equation to represent the Profit: P

• we assume that x = cases of 16 oz cups produced per day and y = cases of 20 oz cups produced per day
• The profit on a case of 20 oz plastic cups is $20 while the profit on a case of 16 oz plastic cups is $25.

Therefore, equation to represent the Profit: ?P=25x+20y?

LPP model is given by ,

max ?P=25x+20y?

subject to,

?x+y=120?

?14x+18y≤1800?

?x=3y?

?x,y≥0?

 What is the maximum profit that can be made given the parameters of this system?

 Equation to represent the Profit: ?P=25x+20y? ........ (1)

we obtained solution as (x,y ) = ( 90 , 30 ) , so we will substitute value of x, y in equation (1)

?P=25x+20y?

?=25∗90+20∗30?

?=2250+600?

= ?2850?

Therefore, the maximum profit that can be made given the parameters of this system is $ 2850

Currently, the smoothie company purchases three times as many 16 oz cups to 20 oz cups,

however, they said in the near future they would like to order four times as many 16 oz cups to 20 oz cups.

How many of each type of cup should be made if the smoothie company changes their order?  

because, of this only one equation will change

equation to represent the number of 16 oz vs 20 oz cups made:

Number of 16 oz vs 20 oz cups made: ?x:y=4:1?

So, ?x=4y?

remaining constraints does not changes

so, LPP model is,

max ?P=25x+20y?

subject to,

?x+y=120?

?14x+18y≤1800?

?x=4y?

?x,y≥0?

we will solve this by substitution,

?x+y=120?

we will substitute ?x=4y? in above equation , we will get

?4y+y=120?

?5y=120?

?y=5120?=24?

put y = 24 in ?x=4y?

we get

x=4y?

x = 4 * 24 = 96

?x=96?

put ?x=96? and y = 24 in ?14x+18y≤1800?

?14x+18y=14∗96+18∗24=1776≤1800?

?14x+18y≤1800? holds for x,y

Therefore, solution for new system is ?(x,y)=(96,24)?

Type of cup should be made if the smoothie company changes their order:

96 cases of 16 oz cups and 24 cases of 20 oz cups produced per day

What would the profit be for this new situation?

as objective function is max ?P=25x+20y?

max ?P=25x+20y?

= ?25∗96+20∗24?

= 2400+ 480

= 2880

Therefore, Profit be for this new situation is $ 2880

(For this new situation : Solution can be verified by drawing graph )

Graph for for this new situation :

PFA