### Video Transcript

A small factory produces two types of metal furniture, π΄ and π΅. They can produce at most 25 pieces of metal furniture in total. The profit from type π΄ is 60 Egyptian pounds and the profit from type π΅ is 40 Egyptian pounds. The factory sells at least two times more of type π΄ than Type π΅. State the objective function and inequalities that will help to find the maximum profit for the factory.

Okay, so in this example, we have a factory that produces two types of furniture, type π΄ and type π΅. Weβre told a bit about what it takes to produce this furniture as well as how much the factory makes when each type is sold. Our question asks us to state the objective function and the inequalities that will help to find the maximum profit for the factory.

We can get started by considering what weβre told about the profit from each type of furniture. Weβre told that for every unit of type π΄ of the furniture sold, the factory makes 60 Egyptian pounds. This means the total profit from this type of furniture will be 60 times π΄. Then weβre told that for each unit of type π΅ sold, the factory makes 40 pounds. The total profit then from type π΅ furniture sales equals 40 times π΅. And the total profit for the factory overall, weβll call it π, equals the sum of these two products.

Because weβre looking to maximize profit in this example, this equation we have for π is our objective function. This is the function we want to maximize given a series of constraints. Constraints have to do with limits on how many units of type π΄ and π΅ we can make. When we think about what the limits are on π΄ and π΅, for one thing, we know that there canβt be a negative number of these types of furniture. In other words, both π΄ and π΅ must be greater than or equal to zero. Along with this, weβre told that the factory can produce at most 25 pieces of metal furniture. This means if we add together the number of type π΄ with type π΅, that total must be less than or equal to 25.

Along with this, weβre told that the factory sells at least two times more of type π΄ than type π΅. This means if we take the number of type π΅ furniture and double that number, then this is less than or equal to the total number of π΄ type sold. Weβve now accounted for all the constraints in this scenario.

As our answer, we can say that our objective function is π equals 60π΄ plus 40π΅. Our inequalities are that π΄ and π΅ are greater than or equal to zero, π΄ plus π΅ is less than or equal to 25, and π΄ is greater than or equal to two times π΅.