To address the problem, we first need to understand and clearly define the constraints involved and then complete the objective function.
### Constraints:
We are given the following constraints:
1. [tex]\( 6x + 2y \geq 720 \)[/tex]
2. [tex]\( -4x + 4y \geq 400 \)[/tex]
Where:
- [tex]\( x \)[/tex] represents the number of cappuccino bottles.
- [tex]\( y \)[/tex] represents the number of latte bottles.
### Simplifying the Constraints:
Let’s simplify these constraints if possible:
#### Constraint 1: [tex]\( 6x + 2y \geq 720 \)[/tex]
We can divide every term by 2:
[tex]\[ 3x + y \geq 360 \][/tex]
#### Constraint 2: [tex]\( -4x + 4y \geq 400 \)[/tex]
We can divide every term by 4:
[tex]\[ -x + y \geq 100 \][/tex]
These simplified constraints are:
1. [tex]\( 3x + y \geq 360 \)[/tex]
2. [tex]\( -x + y \geq 100 \)[/tex]
### Objective Function:
The objective function is given in the form [tex]\( P = Ax + By \)[/tex], where typically [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are coefficients representing the contribution of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to the objective ([tex]\( P \)[/tex]). However, the specific coefficients [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not provided in the problem.
To determine the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] for the objective function, additional context or information about what [tex]\( P \)[/tex] represents (such as profit, cost, etc.) would be necessary. Usually, this function is given based on the problem's goal, such as maximizing profit or minimizing cost.
Since we do not have explicit coefficients in the problem, we denote the objective function in the general form for now:
[tex]\[ P = Ax + By \][/tex]
Thus, the complete answer to the problem as given is:
### Objective Function:
[tex]\[ P = Ax + By \][/tex]
Without specific values for [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we cannot further define this function. However, the constraints have been simplified and stated clearly:
1. [tex]\( 3x + y \geq 360 \)[/tex]
2. [tex]\( -x + y \geq 100 \)[/tex]
To proceed further with solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the next step would involve determining the feasible region from these constraints and optimizing the objective function within that region, given specific values for [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
