A manufacturer produces two types of bottled coffee drinks: cappuccinos and cafés au lait. Each bottle of cappuccino requires 6 ounces of coffee and 2 ounces of milk and earns a profit of . Each bottle of café au lait requires 4 ounces of coffee and 4 ounces of milk and earns a profit of . The manufacturer has 720 ounces of coffee and 400 ounces of milk available for production each day. To meet demand, the manufacturer must produce at least 80 coffee drinks each day. Let the number of cappuccino bottles and the number of café au lait bottles.

Identify the constraints on the system other than and .

DONE

Ask by Chan Long. in the United States

Mar 22,2025

Question

A manufacturer produces two types of bottled coffee drinks: cappuccinos and cafés au lait. Each bottle of cappuccino requires 6 ounces of coffee and 2 ounces of milk and earns a profit of . Each bottle of café au lait requires 4 ounces of coffee and 4 ounces of milk and earns a profit of . The manufacturer has 720 ounces of coffee and 400 ounces of milk available for production each day. To meet demand, the manufacturer must produce at least 80 coffee drinks each day. Let the number of cappuccino bottles and the number of café au lait bottles.

Identify the constraints on the system other than and .

DONE

Ask by Chan Long. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We first identify the use of resources in each drink:

- Each cappuccino (represented by $$ x $$) requires:
  - $$6$$ ounces of coffee
  - $$2$$ ounces of milk

- Each café au lait (represented by $$ y $$) requires:
  - $$4$$ ounces of coffee
  - $$4$$ ounces of milk

The manufacturer has available:
- $$720$$ ounces of coffee
- $$400$$ ounces of milk

In addition, to meet the demand, the manufacturer must produce at least $$80$$ drinks per day. Therefore, we have:

1. **Demand Constraint:**

$$
   x+y \geq 80
   $$

2. **Coffee Usage Constraint:**

Total coffee used is $$6x+4y$$ and it must not exceed $$720$$ ounces:

$$
   6x+4y \leq 720
   $$

3. **Milk Usage Constraint:**

Total milk used is $$2x+4y$$ and it must not exceed $$400$$ ounces:

$$
   2x+4y \leq 400
   $$

Thus, the constraints (other than $$ x \geq 0 $$ and $$ y \geq 0 $$) are:

$$
x+y \geq 80, \quad 6x+4y \leq 720, \quad 2x+4y \leq 400.
$$
The constraints are: - $$ x + y \geq 80 $$ - $$ 6x + 4y \leq 720 $$ - $$ 2x + 4y \leq 400 $$

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