Hello, CMoSeattle!

The problem has too many words . . .

A company produces and sells two brands of pet food: brand X and brand Y.

A case of brand X requires 4 units of nutrient C, 12 units of flavor additive and 8 units of color additive.

A case of brand Y requires 4 units of nutrient C, 6 units of flavor additive and 15 units of color additive.

The supply is limited to: 30 units of nutrient C, 72 units of flavor additive, and 90 units of color additive.

The profit is: $40 per case of brand X, $30 per case of brand Y.

How much of each brand must be produced to maximize profit?

Let = number of cases of brand X. .

Let = number of cases of brand Y. .

We have these facts:

. .

We have these inequalities:

. .

Graph and shade the inequalities.

You should get a pentagon with vertices:

. .

Substitute into the Profit Function,

. . and determine which point produces maximum profit.