Sure! Let's solve the linear programming problem step-by-step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that maximize the given objective function [tex]\( P = 20x + 25y \)[/tex].
### Problem Definition
We need to maximize:
[tex]\[ P = 20x + 25y \][/tex]
Subject to the constraints:
[tex]\[ 0.6x + 1.2y \leq 960 \][/tex]
[tex]\[ 0.03x + 0.04y \leq 36 \][/tex]
[tex]\[ 0.3x + 0.2y \leq 270 \][/tex]
[tex]\[ x \geq 0 \][/tex]
[tex]\[ y \geq 0 \][/tex]
### Step-by-Step Solution
1. Formulating the Objective Function:
We want to maximize:
[tex]\[ P = 20x + 25y \][/tex]
2. Formulating the Constraints:
The given constraints in inequality form are:
1. [tex]\( 0.6x + 1.2y \leq 960 \)[/tex]
2. [tex]\( 0.03x + 0.04y \leq 36 \)[/tex]
3. [tex]\( 0.3x + 0.2y \leq 270 \)[/tex]
4. [tex]\( x \geq 0 \)[/tex]
5. [tex]\( y \geq 0 \)[/tex]
3. Solving the Linear Programming Problem:
Using an appropriate method, such as the Simplex Method or an optimization solver, we solve the linear programming problem. The solution gives us the values of [tex]\( x \)[/tex], [tex]\( y \)[/tex], and the maximum value of [tex]\( P \)[/tex].
4. Interpreting the Solution:
The solution found is:
[tex]\[ x = 600.0 \][/tex]
[tex]\[ y = 450.0 \][/tex]
[tex]\[ P = 23250.0 \][/tex]
This means:
[tex]\[
\begin{align*}
\text{For } x &= 600.0, \\
\text{and } y &= 450.0, \\
\text{the maximum value of } P &= 23250.0.
\end{align*}
\][/tex]
Thus, by following these steps, we determined that the maximum value of the objective function [tex]\( P \)[/tex] is 23,250 when [tex]\( x = 600 \)[/tex] and [tex]\( y = 450 \)[/tex].
