The minimum value of `z=8x+10y` subject to `2x+yge7,2x+3yge15,yge2,xge0,yge0` is

Options

A. 56
B. 52
C. 51
D. 48

Answer

To find the minimum value of \( z = 8x + 10y \) subject to the constraints \( 2x + y \geq 7 \), \( 2x + 3y \geq 15 \), \( y \geq 2 \), \( x \geq 0 \), and \( y \geq 0 \), we can follow these steps:

### Step 1: Convert Inequalities to Equations
First, we convert the inequalities into equations to find the boundary lines:
1. \( 2x + y = 7 \)
2. \( 2x + 3y = 15 \)
3. \( y = 2 \)

### Step 2: Find Intercepts
Next, we find the intercepts for each line:
1. For \( 2x + y = 7 \):
- When \( x = 0 \), \( y = 7 \) (point \( (0, 7) \))
- When \( y = 0 \), \( x = 3.5 \) (point \( (3.5, 0) \))

2. For \( 2x + 3y = 15 \):
- When \( x = 0 \), \( y = 5 \) (point \( (0, 5) \))
- When \( y = 0 \), \( x = 7.5 \) (point \( (7.5, 0) \))

3. For \( y = 2 \):
- This is a horizontal line at \( y = 2 \).

### Step 3: Graph the Lines
Now, we graph the lines on the coordinate plane:
- The line \( 2x + y = 7 \) will intersect the y-axis at \( (0, 7) \) and the x-axis at \( (3.5, 0) \).
- The line \( 2x + 3y = 15 \) will intersect the y-axis at \( (0, 5) \) and the x-axis at \( (7.5, 0) \).
- The line \( y = 2 \) is a horizontal line.

### Step 4: Determine the Feasible Region
The feasible region is where all inequalities overlap. We will shade the regions above each line:
- For \( 2x + y \geq 7 \), shade above the line.
- For \( 2x + 3y \geq 15 \), shade above the line.
- For \( y \geq 2 \), shade above the line.
- For \( x \geq 0 \) and \( y \geq 0 \), we are restricted to the first quadrant.

### Step 5: Identify Corner Points
The corner points of the feasible region can be found by solving the equations:
1. Intersection of \( 2x + y = 7 \) and \( 2x + 3y = 15 \):
- Solve the system of equations:
\[ 2x + y = 7 \quad (1) \]
\[ 2x + 3y = 15 \quad (2) \]
- From (1), \( y = 7 - 2x \).
- Substitute into (2):
\[ 2x + 3(7 - 2x) = 15 \implies 2x + 21 - 6x = 15 \implies -4x = -6 \implies x = 1.5 \]
\[ y = 7 - 2(1.5) = 4 \]
- So, one point is \( (1.5, 4) \).

2. Intersection of \( 2x + y = 7 \) and \( y = 2 \):
- Substitute \( y = 2 \) into \( 2x + y = 7 \):
\[ 2x + 2 = 7 \implies 2x = 5 \implies x = 2.5 \]
- So, another point is \( (2.5, 2) \).

3. Intersection of \( 2x + 3y = 15 \) and \( y = 2 \):
- Substitute \( y = 2 \) into \( 2x + 3y = 15 \):
\[ 2x + 6 = 15 \implies 2x = 9 \implies x = 4.5 \]
- So, another point is \( (4.5, 2) \).

4. The point \( (0, 7) \) is also a corner point.

### Step 6: Evaluate the Objective Function
Now we evaluate \( z = 8x + 10y \) at the corner points:
1. At \( (0, 7) \):
\[ z = 8(0) + 10(7) = 70 \]
2. At \( (1.5, 4) \):
\[ z = 8(1.5) + 10(4) = 12 + 40 = 52 \]
3. At \( (4.5, 2) \):
\[ z = 8(4.5) + 10(2) = 36 + 20 = 56 \]
4. At \( (2.5, 2) \):
\[ z = 8(2.5) + 10(2) = 20 + 20 = 40 \]

### Step 7: Determine the Minimum Value
The minimum value of \( z \) occurs at the point \( (1.5, 4) \) where \( z = 52 \).

### Final Answer
The minimum value of \( z \) is **52**.