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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 . 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 .






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Ask by Chan Long. in the United States
Mar 22,2025

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The constraints are:

Solución

We first identify the use of resources in each drink:
  • Each cappuccino (represented by ) requires:
    • ounces of coffee
    • ounces of milk
  • Each café au lait (represented by ) requires:
    • ounces of coffee
    • ounces of milk
The manufacturer has available:
  • ounces of coffee
  • ounces of milk
In addition, to meet the demand, the manufacturer must produce at least drinks per day. Therefore, we have:
  1. Demand Constraint:
  2. Coffee Usage Constraint:
    Total coffee used is and it must not exceed ounces:
  3. Milk Usage Constraint:
    Total milk used is and it must not exceed ounces:
Thus, the constraints (other than and ) are:

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The Deep Dive

Let’s sip on some fascinating tidbits! The constraints you have come across stem from the resources available and the production requirements. With 720 ounces of coffee and 400 ounces of milk, ensuring you stay within those limits while satisfying the daily demand can be tricky, but it’s essential for running a smooth operation!
If you’re diving deeper into optimizing production, you might enjoy exploring linear programming. It’s a powerful mathematical technique used not only in manufacturing but also in logistics, finance, and even diet planning! Check out some case studies or textbooks on linear optimization for a richer understanding. There’s a world of efficiencies waiting for you to discover!

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