What is Feasible Region?

S3-SA1-0181

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Feasible Region is a specific area on a graph that contains all possible solutions that satisfy a set of given conditions or rules. Think of it as the 'allowed' zone where all your requirements are met at the same time.

Simple Example

Quick Example

Imagine you want to buy samosas and jalebis. You have Rs. 50. One samosa costs Rs. 10 and one jalebi costs Rs. 5. You also want to buy at least 2 samosas. The 'feasible region' would be all the combinations of samosas and jalebis you can buy that fit both your budget and your wish to buy at least 2 samosas.

Worked Example

Step-by-Step

Let's find the feasible region for these conditions: Condition 1: x is greater than or equal to 0 (x >= 0). Condition 2: y is greater than or equal to 0 (y >= 0). Condition 3: x + y is less than or equal to 5 (x + y <= 5).
---Step 1: Draw a graph with x and y axes. Since x >= 0 and y >= 0, our feasible region will only be in the first quadrant (top-right section).
---Step 2: Plot the line for the equation x + y = 5. You can find two points: If x=0, then y=5 (point (0,5)). If y=0, then x=5 (point (5,0)). Draw a straight line connecting these two points.
---Step 3: Now, consider the inequality x + y <= 5. This means the region below or on the line x + y = 5 is part of our solution. Pick a test point like (0,0). 0+0 <= 5 is true, so the region containing (0,0) is the correct side.
---Step 4: Combine all conditions. The region that is in the first quadrant (x>=0, y>=0) AND is below or on the line x + y = 5 is the feasible region.
---Answer: The feasible region is the triangle formed by the points (0,0), (5,0), and (0,5), including its boundaries.

Why It Matters

Understanding feasible regions helps scientists, engineers, and economists make the best decisions when resources are limited. For example, a data scientist uses this to find the best settings for an AI model, or an engineer uses it to design a bridge that is both strong and cost-effective. It helps in optimizing things in various fields.

Common Mistakes

MISTAKE: Not considering all the given conditions when shading the region. | CORRECTION: Carefully check each inequality and make sure your shaded region satisfies ALL of them simultaneously.

MISTAKE: Incorrectly shading the region for an inequality (e.g., shading above the line for 'less than or equal to'). | CORRECTION: Pick a test point (like (0,0) if it's not on the line) and substitute its coordinates into the inequality. If it makes the inequality true, shade the side of the line that contains the test point.

MISTAKE: Drawing a dashed line instead of a solid line for 'greater than or equal to' or 'less than or equal to' inequalities. | CORRECTION: Use a solid line if the inequality includes 'equal to' (>= or <=), as the line itself is part of the feasible region. Use a dashed line only for strict inequalities (> or <).

Practice Questions

Try It Yourself

QUESTION: Shade the feasible region for x >= 2 and y >= 1. | ANSWER: The region to the right of the vertical line x=2 and above the horizontal line y=1, including the lines themselves.

QUESTION: Identify the feasible region for x + y <= 4, x >= 0, y >= 0. | ANSWER: The triangle formed by points (0,0), (4,0), and (0,4), including its boundaries.

QUESTION: A baker wants to make cakes and biscuits. She has 10 kg of flour. Each cake needs 1 kg flour, and each biscuit needs 0.5 kg flour. She wants to make at least 2 cakes. Write the inequalities and describe the feasible region for the number of cakes (x) and biscuits (y). | ANSWER: Inequalities: x >= 2, x >= 0, y >= 0, x + 0.5y <= 10. The feasible region is the area bounded by x=2, y=0, and the line x + 0.5y = 10 (or y = -2x + 20), in the first quadrant.

MCQ

Quick Quiz

Which of these points lies in the feasible region defined by x >= 0, y >= 0, and x + y <= 3?

(4,0)

(0,4)

(1,1)

(-1,2)

The Correct Answer Is:

For (1,1): 1 >= 0 (True), 1 >= 0 (True), 1+1 <= 3 (2 <= 3, True). All conditions are met. Options A and B fail x+y <= 3. Option D fails x >= 0.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. They have conditions like 'deliveries must be within 100 km of the warehouse,' 'each driver can carry max 50 parcels,' and 'deliveries must be completed by 6 PM.' The 'feasible region' helps them find all possible routes and parcel distributions that satisfy all these rules, making their operations efficient.

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, or >=. | CONSTRAINT: A condition or restriction that limits the possible solutions. | OPTIMIZATION: The process of finding the best possible solution under given constraints. | GRAPHING: Representing mathematical relationships visually using coordinates on a plane.

What's Next

What to Learn Next

Now that you understand feasible regions, you can move on to 'Linear Programming.' This exciting concept uses feasible regions to find the absolute best solution (like maximum profit or minimum cost) within all the possible solutions you just learned to identify. Keep practicing!