Certainly! To determine the minimum value of the cost function \( C = 6x + 7y \) subject to the constraints:
1. \( 4x + 3y \geq 24 \)
2. \( x + 3y \geq 15 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)
we follow these steps:
### Step 1: Convert Inequalities to Equalities
First, we need to convert the inequality constraints into system boundaries that we can graph or analyze.
1. \( 4x + 3y \geq 24 \) can be rewritten as \( 4x + 3y = 24 \).
2. \( x + 3y \geq 15 \) can be rewritten as \( x + 3y = 15 \).
### Step 2: Calculate Intersection Points
Now we find the points of intersection of the constraints \( 4x + 3y = 24 \) and \( x + 3y = 15 \), and also check boundaries \( x = 0 \) and \( y = 0 \).
Set up the equations as follows and solve them:
[tex]\[
\begin{cases}
4x + 3y = 24 \\
x + 3y = 15
\end{cases}
\][/tex]
Subtract the second equation from the first one:
[tex]\[
(4x + 3y) - (x + 3y) = 24 - 15 \\
3x = 9 \\
x = 3
\][/tex]
Substitute \( x = 3 \) into the second equation:
[tex]\[
3 + 3y = 15 \\
3y = 12 \\
y = 4
\][/tex]
So the intersection point of \( 4x + 3y = 24 \) and \( x + 3y = 15 \) is \( (3, 4) \).
### Step 3: Evaluate the Objective Function at the Vertex
We check this feasible point against the objective function \( C = 6x + 7y \):
[tex]\[
C = 6(3) + 7(4) \\
C = 18 + 28 \\
C = 46
\][/tex]
Thus, the minimum value of \( C = 46 \) occurs at \( x = 3 \) and \( y = 4 \).
### Final Answer
The minimum value of the cost function \( C = 6x + 7y \) subject to the given constraints is:
[tex]\[
C = 46
\][/tex]
with [tex]\( x = 3 \)[/tex] and [tex]\( y = 4 \)[/tex].
