A food factory is making a beverage for a customer from mixing two different existing
products A and B. The compositions of A and B and prices ($/L) are given as follows,
Amount.(L) in/100L of A and B

{Lime,Orange,Mango,Cost($/L)}; {A,2,6,4,5}; {B,7,4,8,15}. 
The customer requires that there must be atleast 5 Litres(L)Orange and atleast 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage.The customer needs atleast 150 Litres of the beverage per week.

We are tasked with creating a beverage that blends products A and B to meet certain concentration and cost requirements.

Let’s define the following variables:

- $ x $ = Litres of product A per 100 litres of the beverage
- $ y $ = Litres of product B per 100 litres of the beverage

### Objective:
We aim to minimize the total cost of the mixture while satisfying the customer's requirements. This is a standard linear programming problem, and the formulation is as follows:

### Cost Function to Minimize:
The cost of the mixture, in terms of the components A and B, is given by:
\[
\text{Cost} = 5x + 15y
\]

### Subject to the following constraints:

1. **Orange content**: The customer requires at least 5 litres of orange per 100 litres of the beverage. Since A has 6 litres of orange per 100 litres and B has 4 litres, we get the constraint:
\[
6x + 4y \geq 5
\]

2. **Mango content**: The customer requires at least 5 litres of mango per 100 litres of the beverage. Since A has 4 litres of mango per 100 litres and B has 8 litres, we get the constraint:
\[
4x + 8y \geq 5
\]

3. **Lime content**: The customer requires no more than 6 litres of lime per 100 litres of the beverage. Since A has 2 litres of lime per 100 litres and B has 7 litres, we get the constraint:
\[
2x + 7y \leq 6
\]

4. **Non-negative quantities**: The quantities of A and B must be non-negative:
\[
x \geq 0, \quad y \geq 0
\]

5. **Total volume constraint**: The customer needs at least 150 litres of the beverage per week. Assuming we solve the problem for a 100-litre batch, this is satisfied implicitly by the linear model.

### Linear Programming Problem:

Minimize:
\[
5x + 15y
\]

Subject to:
\[
\begin{align*}
6x + 4y &\geq 5 \\
4x + 8y &\geq 5 \\
2x + 7y &\leq 6 \\
x &\geq 0 \\
y &\geq 0
\end{align*}
\]

We can solve this system to find the optimal values for $x$ and $y$, which will give us the proportions of A and B in the beverage that minimize cost while satisfying the customer's requirements.

Would you like me to solve this optimization problem and provide the values for $x$ and $y$? Let me know if you want more details or if you have any questions.

---

Here are 5 related questions for further understanding:
1. What method can be used to solve this kind of linear programming problem?
2. How do constraints in a linear programming problem influence the solution?
3. How would changing the cost of product A or B affect the solution?
4. Can you have a situation where no solution satisfies all constraints?
5. How can graphical methods help visualize this optimization problem?

**Tip:** When solving linear programming problems, checking the feasibility of each constraint helps in understanding the problem structure.

A food factory is making a beverage for a customer from mixing two different existing products A and B. The compositions of A and B and prices ($/L) are given as follows, Amount.(L) in/100L of A and B: {Lime, Orange, Mango, Cost($/L)}; {A, 2, 6, 4, 5}; {B, 7, 4, 8, 15}. The customer requires that there must be at least 5 Litres (L) of Orange and at least 5 Litres of Mango concentrate per 100 Litres of the beverage respectively, but no more than 6 Litres of Lime concentrate per 100 Litres of beverage. The customer needs at least 150 Litres of the beverage per week.

Solve a linear programming problem to minimize the cost of a beverage made by mixing two products while satisfying ingredient constraints. Learn how to apply cost minimization and inequality constraints for lime, orange, and mango content. Suitable for students in Grades 11-12 or undergraduates.

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