Math Snap

STEP 1

1. The problem is a linear programming problem.
2. The objective function to minimize is $z = 700x + 600y$.
3. The feasible region is defined by the vertices: $(5, 45)$, $(40, 45)$, $(30, 40)$, $(35, 25)$, and $(40, 5)$.

STEP 2

1. Identify the vertices of the feasible region.
2. Evaluate the objective function at each vertex.
3. Determine the minimum value of the objective function.

STEP 3

Identify the vertices of the feasible region:
The vertices are given as:
- $(5, 45)$
- $(40, 45)$
- $(30, 40)$
- $(35, 25)$
- $(40, 5)$

STEP 4

Evaluate the objective function $z = 700x + 600y$ at each vertex:
1. At $(5, 45)$:
$$ z = 700(5) + 600(45) = 3500 + 27000 = 30500
\] 2. At $(40, 45)$:
$$ z = 700(40) + 600(45) = 28000 + 27000 = 55000
\] 3. At $(30, 40)$:
$$ z = 700(30) + 600(40) = 21000 + 24000 = 45000
\] 4. At $(35, 25)$:
$$ z = 700(35) + 600(25) = 24500 + 15000 = 39500
\] 5. At $(40, 5)$:
$$ z = 700(40) + 600(5) = 28000 + 3000 = 31000
\]

Determine the minimum value of the objective function:
The calculated values are:
- $z = 30500$ at $(5, 45)$
- $z = 55000$ at $(40, 45)$
- $z = 45000$ at $(30, 40)$
- $z = 39500$ at $(35, 25)$
- $z = 31000$ at $(40, 5)$
The minimum value is $\boxed{30500}$ at the vertex $(5, 45)$.