College

Graph the feasible region subject to the following constraints:

[tex]
\begin{array}{c}
x + y \leq 20000 \\
x \geq 3000 \\
y \geq 4000
\end{array}
[/tex]

Answer :

To graph the feasible region described by the constraints [tex]\( x \cdot y \leq 20000 \)[/tex], [tex]\( x \geq 3000 \)[/tex], and [tex]\( y \geq 4000 \)[/tex], follow these steps:

1. Understand Each Constraint:

- [tex]\( x \cdot y \leq 20000 \)[/tex]: This represents a hyperbola. The product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] should not exceed 20,000.
- [tex]\( x \geq 3000 \)[/tex]: This is a vertical line at [tex]\( x = 3000 \)[/tex]. We are considering the region to the right of this line.
- [tex]\( y \geq 4000 \)[/tex]: This is a horizontal line at [tex]\( y = 4000 \)[/tex]. We are considering the region above this line.

2. Plot the Boundary Curves and Lines:

- Hyperbola (Boundary of [tex]\( x \cdot y = 20000 \)[/tex]):
- Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]: [tex]\( y = \frac{20000}{x} \)[/tex].
- This curve will go through points like [tex]\( (4000, 5) \)[/tex] and [tex]\( (5000, 4) \)[/tex], among others.
- Vertical Line: Plot the line [tex]\( x = 3000 \)[/tex].
- Horizontal Line: Plot the line [tex]\( y = 4000 \)[/tex].

3. Identify the Feasible Region:

- The region of interest is where all the constraints are satisfied simultaneously:
- It must lie below the hyperbola, where [tex]\( y \leq \frac{20000}{x} \)[/tex].
- It must be to the right of [tex]\( x = 3000 \)[/tex].
- It must be above [tex]\( y = 4000 \)[/tex].

4. Sketch the Feasible Region:

- Begin by drawing the hyperbola curve [tex]\( y = \frac{20000}{x} \)[/tex], but only consider the part where [tex]\( x \)[/tex] is more than 3000 to ensure it satisfies [tex]\( x \geq 3000 \)[/tex].
- Draw the vertical line at [tex]\( x = 3000 \)[/tex] and shade the region to the right.
- Draw the horizontal line at [tex]\( y = 4000 \)[/tex] and shade the region above it.
- The feasible region is where all these shaded parts overlap, typically a right-bottom triangular or quadrilateral region for this specific setup.

5. Visualizing and Conclusion:

- The feasible region will be bounded by the hyperbola and lie entirely in the first quadrant with [tex]\( x \geq 3000 \)[/tex] and [tex]\( y \geq 4000 \)[/tex].
- Note any intersection points to more clearly define the bounds of the region. For example, solve the equation [tex]\( 3000 \times y = 20000 \)[/tex] to find where the vertical line intersects the hyperbola if necessary.

By following these steps, you should be able to visualize and sketch the feasible region that satisfies all given constraints.