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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 [tex]$\$[/tex]0.40[tex]$. Each bottle of café au lait requires 4 ounces of coffee and 4 ounces of milk and earns a profit of $[/tex]\[tex]$0.50$[/tex]. 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 [tex]$x$[/tex] be the number of cappuccino bottles and [tex]$y$[/tex] be the number of café au lait bottles.

Identify the constraints on the system other than [tex]$x \geq 0$[/tex] and [tex]$y \geq 0$[/tex]:

1. [tex]$x + y \geq 80$[/tex]
2. [tex]$6x + 4y \leq 720$[/tex]
3. [tex]$2x + 4y \leq 400$[/tex]



Answer :

GinnyAnswer
Certainly! Let's identify the necessary constraints for the manufacturer's production process.

We are given the following information:
- Each bottle of cappuccino (let's call it [tex]\( x \)[/tex]) requires 6 ounces of coffee and 2 ounces of milk, earning a profit of [tex]$\$[/tex]0.40[tex]$. - Each bottle of café au lait (let's call it \( y \)) requires 4 ounces of coffee and 4 ounces of milk, earning a profit of $[/tex]\[tex]$0.50$[/tex].
- The manufacturer has 720 ounces of coffee and 400 ounces of milk available daily.
- The manufacturer needs to produce at least 80 bottles of coffee drinks daily, combining both types.

Let's summarize these conditions into constraints:

1. Minimum Production Requirement: The manufacturer must produce at least 80 bottles (cappuccino and café au lait combined) each day.
[tex]\[ x + y \geq 80 \][/tex]

2. Coffee Supply Constraint: The total amount of coffee used by cappuccinos and cafés au lait should not exceed 720 ounces.
[tex]\[ 6x + 4y \leq 720 \][/tex]

3. Milk Supply Constraint: The total amount of milk used by cappuccinos and cafés au lait should not exceed 400 ounces.
[tex]\[ 2x + 4y \leq 400 \][/tex]

Now, assessing the given options:

- [tex]\( x + y \geq 80 \)[/tex]: This matches the minimum production requirement constraint.
- [tex]\( 0.4x + 0.5y \geq 100 \)[/tex]: This is not a valid constraint derived from the given information.
- [tex]\( 6x + 4y \leq 720 \)[/tex]: This matches the coffee supply constraint.
- [tex]\( 2x + 4y \leq 400 \)[/tex]: This matches the milk supply constraint.
- [tex]\( 6x + 2y \geq 720 \)[/tex]: This is not correct given the resource limitations.
- [tex]\( 4x + 4y \geq 400 \)[/tex]: This is not correct given the resource limitations.

Therefore, the correct constraints on the system, other than [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex], are:

[tex]\[ x + y \geq 80 \][/tex]

[tex]\[ 6x + 4y \leq 720 \][/tex]

[tex]\[ 2x + 4y \leq 400 \][/tex]

These constraints ensure that the manufacturer meets the demand and stays within the available resources.

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