Succulent Citrus produces orange juice and orange concentrate. This year the company anticipates a demand of at least 11,000 quarts of orange juice and 1,500 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 260,000 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?

This is a problem of optimization, therefore we have to identify the constraints and the objective function. First, we are gonna name the variables of the problem as follows:

[tex]Q_{O}[/tex]: quarts of orange juice produced

[tex]Q_{C}[/tex]: quarts of orange concentrate produced

The first and second constraints are related with the demand, both quantities of orange juice and concentrate must be greater or equal than the given values for demand:

[tex]Q_{O}\geq11000 \\Q_{C}\geq 1500[/tex]

The third constraint is related with the oranges required, as the company anticipates using at least 260000 oranges for these products, the sum of the oranges used to produce both products must be greater or equal than this value:

[tex]10Q_{O}+50Q_{C}\geq 260000[/tex]

And finally the objective function is the equation to determine the costs:

[tex]C(Q)=0.5Q_{O}+2Q_{C}[/tex]

We look for minimizing this last function, therefore we use the Solver tool in an Excel spreadsheet (file attached), and we add the constraints explained above. Finally we get:

[tex]Q_{O}= 11,000 quarts\\Q_{C}= 3,000 quarts\\C =[/tex] $11,500