Succulent Citrus produces orange juice and orange concentrate. This year the company anticipates a demand of at least 13,000 quarts of orange juice and 500 quarts of orange concentrate. Each quart of orange juice requires 10 oranges, and each quart of concentrate requires 50 oranges. The company also anticipates using at least 180,000 oranges for these products. Each quart of orange juice costs the company 50¢ to produce, and each quart of concentrate costs $2.00 to produce. How many quarts of each product should Succulent Citrus produce to meet the demand and minimize total costs?

Answer: To minimize cost and meet demand, the company has to produce 15,000 quarts of orange juice which will cost $7,500 and 600 quarts of orange concentrate which will cost $1,200. Therefore a total of $8,700 will be needed for the production.

Step-by-step explanation:

Since the company wants to produce at least 13,000 orange juice and 500 quarts of oranges concentration,

10 oranges are used to produce each orange juice, and 50 oranges are used to produce each quarts of oranges concentration.

Therefore; how many oranges are needed in total.

13000 * 10 = 130,000 orange

500 * 50 = 25,000 oranges

Total oranges = 130,000 + 25,000 = 155,000 oranges

Since the company estimate to use at least 180,000 oranges and 155,000 has been used to meet the minimum required production of each product.

Therefore the remaining oranges are;

180,000 - 155,000 = 25,000

The question now becomes; what product are we to produce with 25,000 oranges, since the cost of producing each orange juice and orange concentration are 50 cent and $2.00 respectively.

Since the company wants to minimize cost, their have to use most of the remaining oranges to produce orange juice;

If we estimate the company to use 20,000 oranges to produce oranges juice,

20,000 : 10 = 2000 orange juice

The 2000 orange juice will cost $1,000 to produce.

25,000 - 20,000 = 5,000 oranges

If we use the remaining 5,000 oranges to produce quarts of concentrate;

5,000 : 50 = 100 quarts of oranges concentrate

The 100 orange concentrate will cost $200.

Therefore the company needs to produce

13,000 + 2,000 = 15,000 quarts of orange juice

500 + 100 = 600 quarts of orange concentrate.

To minimize cost and meet demand, more oranges juice has to be produced, than orange concentrate, because the initial proportion was 13,000:500

But since the company has more oranges, it has to invest more in production of the juice.