How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs?

Question

Fancy Pineapple produces pineapple juice and canned pineapple rings. This year the company anticipates a demand of at least 10,000 pints of pineapple juice and 1,000 cans of pineapple rings. Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple. The company anticipates using at least 20,000 pineapples for these products. Each pint of pineapple juice costs the company 20 cents to produce, and each can of pineapple rings costs 50 cents to produce. How many pints of pineapple juice and cans of pineapple rings should Fancy Pineapple produce to meet the demand and minimize total costs?

Number of pints of Pineapple Juice to Produce =

Number of Cans of Pineapple Rings to Produce =

Total Cost =

Answer 1

2x + y <= 20000

x>=10000 y>=1000

.20x + .50y =Cost(x,y)

The vertices of the critical region are

(0,1000)

(500,1000)

(1000,0)

(1000,0) minimizes the objective cost function at $200

Answer 2

Hello,

With this problem, I would use system of equations in order to define the parameters for the problem. This will probably work best with graph paper rather than just problems, but I will do my best to share my thought process and my final conclusions here.

Let's break the problem down into multiple equations.

We will let X represent the pineapple juice and Y represents the pineapple rings.

First, I take the first line and create two limits for my graphs.

This year the company anticipates a demand of at least 10,000 pints of pineapple juice and 1,000 cans of pineapple rings.

Since the company anticipates at least, then our amounts of products have to be greater than the minimums set here.

X >/= 10,000

Y >/= 1,000

Now. let's look at the next part of our word problem:

Each pint of pineapple juice requires 2 pineapples, and each can of pineapple rings requires 1 pineapple.

So in order to get one item, we need the set amount of pineapples to make it. This means that,

1 can of pineapple juice = 2X

1 can of pineapple rings = Y ( or 1Y)

Now, take the next line of the problem:

The company anticipates using at least 20,000 pineapples for these products.

This means that we can create the equation:

2X + Y >/= 20,000 (This represents the total amount of pineapples used by the company)

Now for our last parameter:

Each pint of pineapple juice costs the company 20 cents to produce, and each can of pineapple rings costs 50 cents to produce.

0.20X = Pineapple juice

0.50Y = Pineapple rings

0.2X + 0.5Y = total cost

(These numbers will be used when calculation costs)

Okay, Now is when I would want to use graph paper or a calculator to show the rest of the work.

I know that X will be greater than 10,000 and Y will be greater than 1,000. So one coordinate for my graph will be

(10000, 1000)

Now for my other limits, I will convert 2X + Y >/= 20,000 into a linear equation to graph.

Y = 20000-2X

To create graph points, I will different coordinates.

Y = 20000-2 (10000)

Y= 0

(10000, 0)

1000 = 20000-2X

-19000 = -2X

9500= X

(9500, 1000)

(After putting points on my graph, I have decided that my lowest point that will meet all three criteria is to use my original point of (10000, 1000) because that meets the minimum requirements of the problem and will use more than the minimum number of pineapples. )

Now to calculate cost:

0.2X + 0.5Y = total cost

0.2 (10000) + 0.5 (1000) = Cost

2000 + 500 = Cost

2500= cost

Number of pints of Pineapple Juice to Produce =10,000

Number of Cans of Pineapple Rings to Produce =1,000

Total Cost =$2500

This is the point on the graph where all required conditions have been met with the first two equations.