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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.
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.
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