To solve the given linear programming problem using the method of linear programming, we need to follow these steps:
1. Formulate the Objective Function:
The goal is to minimize the objective function [tex]\( C = x + 2y \)[/tex].
2. Identify the Constraints:
The problem includes the following constraints:
[tex]\[
\begin{cases}
4x + 7y \leq 48 \\
2x + y = 22 \\
x \geq 0 \\
y \geq 0
\end{cases}
\][/tex]
3. Graphical Representation:
Since the problem involves two variables, we can visualize the constraints graphically to understand the feasible region.
- Plot the inequality [tex]\( 4x + 7y \leq 48 \)[/tex].
- Plot the equality [tex]\( 2x + y = 22 \)[/tex].
The feasible region is where all constraints overlap in the first quadrant because [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex].
4. Find the Intersection Points:
To locate the optimal point, we need to find the intersection points of the constraints.
- Solve the system of equations [tex]\( 2x + y = 22 \)[/tex] and [tex]\( 4x + 7y = 48 \)[/tex].
From [tex]\( 2x + y = 22 \)[/tex]:
[tex]\[ y = 22 - 2x \][/tex]
Substitute [tex]\( y = 22 - 2x \)[/tex] into [tex]\( 4x + 7y = 48 \)[/tex]:
[tex]\[ 4x + 7(22 - 2x) = 48 \][/tex]
[tex]\[ 4x + 154 - 14x = 48 \][/tex]
[tex]\[ -10x + 154 = 48 \][/tex]
[tex]\[ -10x = 48 - 154 \][/tex]
[tex]\[ -10x = -106 \][/tex]
[tex]\[ x = 10.6 \][/tex]
Substitute [tex]\( x = 10.6 \)[/tex] back into [tex]\( y = 22 - 2x \)[/tex]:
[tex]\[ y = 22 - 2(10.6) \][/tex]
[tex]\[ y = 22 - 21.2 \][/tex]
[tex]\[ y = 0.8 \][/tex]
5. Evaluate the Objective Function:
Substitute the coordinates [tex]\( (10.6, 0.8) \)[/tex] into the objective function [tex]\( C = x + 2y \)[/tex]:
[tex]\[ C = 10.6 + 2(0.8) \][/tex]
[tex]\[ C = 10.6 + 1.6 \][/tex]
[tex]\[ C = 12.2 \][/tex]
Therefore, the minimum value of the objective function [tex]\( C \)[/tex] is [tex]\( 12.2 \)[/tex], which occurs at [tex]\( x = 10.6 \)[/tex] and [tex]\( y = 0.8 \)[/tex].
To summarize:
- The optimal solution is [tex]\( x = 10.6 \)[/tex] and [tex]\( y = 0.8 \)[/tex].
- The minimum value of the objective function [tex]\( C = x + 2y \)[/tex] is [tex]\( 12.2 \)[/tex].
