# 10.6.3. Linear Programing and Optimization

A baby food factory produces two types of food with specific nutrition values. The first type, denoted by , whose cost is LE, contains 3 units of vitamin A and 4 of vitamin B, and the second type, denoted by , whose cost is LE, contains 4 units of vitamin A and 3 of vitamin B. If a child needs at least 140 units of vitamin A and 100 units of vitamin B 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, , , ,

### Example

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.

### Solution

Suppose is the number of the first type of baby food, and is the number of the second type.

Since it is impossible to purchase a negative number of the first type or the second type, both and must be greater than or equal to 0.

When referring to vitamin A, the words "at least" indicate that the sum of and is greater than or equal to 140, and when referring to vitamin B, the words "at least" indicate that the sum of and is greater than or equal to 100.

Since the price of the first type is LE, the total price of all of the first type purchased is LE, and since the price of the second type is LE, the total price of all of the second type purchased is LE. Therefore, the total price, , of all the baby food purchased is LE.

This means that the relations needed to determine the quantity to be purchased of each type at the lowest possible cost are , , , , .

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