What is a Feasible Region?

S3-SA1-0814

Grade Level:

Class 6

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

Definition

What is it?

A Feasible Region is a specific area or space on a graph that shows all possible solutions to a problem. Think of it as the 'allowed' zone where all the rules or conditions of a problem are met at the same time. Any point inside this region is a valid solution.

Simple Example

Quick Example

Imagine you have 20 rupees and want to buy lollipops (costing 5 rupees each) and chocolates (costing 10 rupees each). You can't spend more than 20 rupees. The 'feasible region' would be all the combinations of lollipops and chocolates you can buy without exceeding 20 rupees.

Worked Example

Step-by-Step

Let's say you have 10 hours to study for two subjects: Math and Science. You want to study Math for at least 3 hours and Science for at least 2 hours. Also, you can't study more than 10 hours total. What is your feasible region for study times?

1. Let 'M' be the hours you study Math and 'S' be the hours you study Science.
---2. Rule 1: You study Math for at least 3 hours. So, M >= 3.
---3. Rule 2: You study Science for at least 2 hours. So, S >= 2.
---4. Rule 3: Total study time cannot be more than 10 hours. So, M + S <= 10.
---5. Rule 4: You cannot study for negative hours. So, M >= 0 and S >= 0 (though M>=3 and S>=2 already cover this).
---6. If you plot these rules on a graph (M on x-axis, S on y-axis), the 'feasible region' is the triangular area where M >= 3, S >= 2, and M + S <= 10 all overlap.
---7. For example, studying Math for 4 hours and Science for 5 hours (total 9 hours) is inside the feasible region. Studying Math for 2 hours and Science for 2 hours (total 4 hours) is NOT feasible because M is not >= 3.
---8. The feasible region is the set of all (M, S) pairs that satisfy M >= 3, S >= 2, and M + S <= 10.

Why It Matters

Understanding feasible regions helps scientists, engineers, and economists find the best possible solutions to complex problems. For instance, a data scientist uses this to optimize how much memory a computer program should use, or a physicist might use it to find the best conditions for an experiment. It helps make smart decisions in many real-world situations.

Common Mistakes

MISTAKE: Only considering one rule at a time when finding the region. | CORRECTION: The feasible region must satisfy ALL the given rules or conditions simultaneously. It's the area where all conditions overlap.

MISTAKE: Confusing 'greater than or equal to' (>=) with 'less than or equal to' (<=) when shading the region on a graph. | CORRECTION: Always double-check the inequality sign. For '>=', you shade above or to the right of the line. For '<=', you shade below or to the left.

MISTAKE: Forgetting that variables like time or quantity cannot be negative, leading to regions extending into negative axes. | CORRECTION: Always remember that in most real-world problems, quantities like number of items, hours, or distance cannot be negative, so the feasible region is usually restricted to the first quadrant (where both x and y are positive).

Practice Questions

Try It Yourself

QUESTION: You want to buy pens (x) and notebooks (y). You must buy at least 3 pens. Write this as an inequality. | ANSWER: x >= 3

QUESTION: A baker needs to make cakes (C) and pastries (P). He has enough ingredients for a maximum of 10 items in total. Write this as an inequality. | ANSWER: C + P <= 10

QUESTION: Your phone battery (B) should be charged for at least 20 minutes but not more than 60 minutes. Write two inequalities to represent this. | ANSWER: B >= 20 and B <= 60

MCQ

Quick Quiz

Which of the following describes a Feasible Region?

Any area on a graph

The area where only one condition is met

The area where all conditions of a problem are met simultaneously

The area outside all conditions

The Correct Answer Is:

A feasible region is specifically the area where all the given constraints or conditions of a problem overlap and are satisfied. It's not just any area, nor is it where only one condition is met, or outside the conditions.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express planning delivery routes. They have rules: each truck can carry a certain weight, must deliver within a time limit, and drivers have maximum driving hours. The 'feasible region' helps them find all possible routes that follow these rules, allowing them to choose the most efficient one. This is called optimization!

Key Vocabulary

Key Terms

INEQUALITY: A mathematical statement showing that two values are not equal, using symbols like <, >, <=, >=. | CONSTRAINT: A rule or condition that limits what can be done in a problem. | OPTIMIZATION: The process of finding the best possible solution among all feasible ones. | GRAPH: A diagram showing the relationship between two or more variables.

What's Next

What to Learn Next

Now that you understand what a Feasible Region is, the next step is to learn about 'Linear Programming'. This builds on feasible regions by helping you find the 'best' point within that region, like finding the maximum profit or minimum cost. Keep up the great work!