# 10.6.3. Linear Programing and Optimization

Using linear programming, find the minimum and maximum values of the function taking in consideration that , , , and .

• Athe minimum value is 0, the maximum value is 9
• Bthe minimum value is , the maximum value is 1
• Cthe minimum value is , the maximum value is
• Dthe minimum value is , the maximum value is 1
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Given the graph below, and that , , , and , determine at which point the function has its maximum using the linear programming.

• A
• B
• C
• D
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A seafood restaurant serves two different kinds of fish; cod and eel. The restaurant needs not more than 75 fishes every day, as it doesn't serve more than 35 cods and 40 eels a day. The price of a cod is LE and that of an eel is LE. If represents the amount of cod, represents the amount of eel, and represents the total price of the fish, state the objective function.

• A
• B
• C
• D
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Determine the values of and that maximize the function .

• A
• B
• C
• D
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A seafood restaurant sells two types of cooked fish; cod and eel. The restaurant sells NOT less than 40 fish every day. It doesn't consume more than 30 cods and NOT more than 45 eels. The price of a cod is LE and that of an eel is LE. If represents the amount of cod, represents the amount of eel, and represents the total price of fish, determine the relations that will help the restaurant manager in this situation.

• A, , , , ,
• B, , , , ,
• C, , , , ,
• D, , , , ,
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A candy shop sells marshmallows and cola candies; the price of a marshmallows bag is LE, and that of a cola candy bag is LE. Determine the least cost at which the kid can buy from the 2 types using the figure below showing the restrictions of the situation, knowing that represents the amount of marshmallow bags, and represents that of cola candy bags.

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A baby food factory produces two different types of food jars with different nutritional values. The first type has 4 units of vitamin A and 2 of vitamin B, while the second has 2 units of vitamin A and 3 of vitamin B. Every child requires at least 120 units of vitamin A and 100 units of vitamin B per meal. The cost of the first type is LE, while that of the second is LE. Using the graph below, determine the number of jars to be consumed by a child that supplies the needed nutrients per meal at the lowest cost possible.

• Ajars of the first type , jars of the second type
• Bjars of the first type , jars of the second type
• Cjars of the first type , jars of the second type
• Djars of the first type , jars of the second type
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A baby food factory produces two different types of food jars with different nutritional values. The first type has 2 units of vitamin A and 4 units of vitamin B, while the second one has 4 units of vitamin A and 2 units of vitamin B. Every child requires at least 100 units of vitamin A and 140 units of vitamin B in each meal. The cost of the first type is LE, while that of the second is LE. Using the graph below, determine the objective function and the lowest possible cost to supply a child with the required nutrients.

• A, and the lowest possible cost is LE.
• B, and the lowest possible cost is LE.
• C, and the lowest possible cost is LE.
• D, and the lowest possible cost is LE.
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A baby food factory produces two types of food with specific nutrition values. The first type, whose cost is LE, contains 3 units of vitamin and 4 of vitamin , and the second type, whose cost is LE, contains 4 units of vitamin and 3 of vitamin . If a child needs at least 140 units of vitamin and 100 units of vitamin to satisfy his nutrition needs per meal, state the equations needed to determine the quantity to be purchased from each type at the lowest possible cost.

• A, , , ,
• B, , , ,
• C, , , ,
• D, , , ,
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A workshop of two workers produces two types of iron desks; one of the workers builds the desks and the other one sprays them. It takes the first worker 4 hours to build a unit of the first type and 3 hours to build a unit of the second type, while it takes the second worker 3 hours to spray a unit of the first type and 4 hours to spray a unit of the second type. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of LE from each single unit, determine the relations required for calculating the number of units of the first type and the number of units of the second type to be produced every day from each type to maximize the profit .

• A, , , ,
• B, , , ,
• C, , , ,
• D, , , ,
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A factory produces two types of iron sheet offices. One of the workers builds the desks and another one sprays them. It takes the first worker 3.5 hours to build a unit of the first type and 2 hours to build a unit of the second type, while it takes the second worker 4 hours to spray a unit of the first type and 2 hours to spray a unit of the second type. The first worker works at least 5 hours a day and the other one works a maximum of 8 hours a day. If the factory earns a profit of LE from each single unit, determine the number of units to be produced every day from each type in order to maximize the profit.

• A first type: 0 units, second type: 2 units
• B first type: 0 units, second type: 4 units
• C first type: 2 units, second type: 0 units
• D first type: 4 units, second type: 0 units
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Two packages of food supplies are available; the first gives 4 calories and has 6 units of vitamin C, and the second gives 3 calories and has 4 units of vitamin C. We need at least 37 calories and 22 units of vitamin C. Given that the prices of the first and the second are LE and LE respectively, state the objective function used to determine the minimum cost of buying the needed nutrients.

• A
• B
• C
• D
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Find the greatest possible value of the objective function expressed by knowing that , , , , and .

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A farmer found that he can improve the quality of his produce if he uses at least 18 units of nitrogen-based Compounds and 6 units of phosphate Compounds for each hectare. There are two types of fertilizers; and . The contents and cost of each are shown in the following table:

The FertilizerNumber of Units of Nitrogen-Based Compounds per kgNumber of Units of Phosphate Compounds per kgCost for each kg (LE)
A32170
B61120

Given that the following figure represents this situation, find the least cost of combination of fertilizers and so that the farmer can provide a sufficient number of units of both compounds to improve the quality of his produce.

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