Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S3-SA5-0142
What is the Concept of Optimisation Graphically?
Grade Level:
Class 10
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Graphically, optimisation is finding the highest or lowest point (maximum or minimum) on a graph that represents a problem. It helps us find the best possible outcome, like the most profit or the least cost, by looking at how different choices change the result.
Simple Example
Quick Example
Imagine you're selling samosas and you want to earn the most money. If you graph 'number of samosas sold' vs. 'profit earned', optimisation is about finding the point on that graph where your profit is highest. It helps you decide the 'best' number of samosas to sell.
Worked Example
Step-by-Step
Let's say a farmer wants to build a rectangular fence using 20 meters of wire. We want to find the maximum area he can enclose.
1. Let the length be 'L' and the width be 'W'. The perimeter is 2L + 2W = 20 meters. So, L + W = 10.
---2. We can write W = 10 - L. The area 'A' is L * W.
---3. Substitute W into the area formula: A = L * (10 - L) = 10L - L^2.
---4. This is a quadratic equation, which forms a parabola when graphed. Since the L^2 term is negative, the parabola opens downwards, meaning it has a maximum point.
---5. The x-coordinate (here, L) of the vertex of a parabola y = ax^2 + bx + c is given by -b/(2a). For A = -L^2 + 10L, a = -1 and b = 10.
---6. So, L = -10 / (2 * -1) = -10 / -2 = 5 meters.
---7. If L = 5 meters, then W = 10 - 5 = 5 meters.
---8. The maximum area is A = 5 * 5 = 25 square meters.
Answer: The maximum area the farmer can enclose is 25 square meters when both length and width are 5 meters.
Why It Matters
Optimisation is super important for making smart decisions in many fields. Engineers use it to design cars that use less fuel, while data scientists use it to make AI models learn faster. It's key to solving problems efficiently, from planning city traffic to managing a startup's finances.
Common Mistakes
MISTAKE: Confusing local maximum/minimum with global maximum/minimum on a complex graph. | CORRECTION: Always check the entire domain of the graph to find the absolute highest or lowest point, not just a peak or valley in one section.
MISTAKE: Assuming that a graph always has a maximum or minimum within a given range. | CORRECTION: Some graphs might keep increasing or decreasing without ever reaching a peak or valley, or the optimal point might be at the boundary of the allowed range.
MISTAKE: Not understanding what the axes represent in an optimisation problem. | CORRECTION: Clearly identify what the x-axis (input/decision variable) and y-axis (output/objective function) stand for to correctly interpret the graph's optimal points.
Practice Questions
Try It Yourself
QUESTION: A small shop's daily profit (in Rupees) can be modelled by P = -x^2 + 12x - 20, where 'x' is the number of items sold. What is the number of items 'x' that will give the maximum profit? | ANSWER: x = 6 items
QUESTION: If the cost of producing 'n' units of a product is given by C(n) = n^2 - 10n + 30, what is the minimum cost of production? | ANSWER: Minimum cost = 5 units, Cost = 5 Rupees
QUESTION: A cricket ball is hit, and its height 'h' (in meters) after 't' seconds is given by h(t) = -5t^2 + 20t + 1. Graph this function for t from 0 to 4 seconds. What is the maximum height the ball reaches and at what time 't'? | ANSWER: Maximum height = 21 meters, at t = 2 seconds
MCQ
Quick Quiz
Which of the following best describes the goal of graphical optimisation?
To draw a neat and colourful graph.
To find the highest or lowest value of a function represented on a graph.
To understand the slope of a line.
To calculate the area under a curve.
The Correct Answer Is:
B
Graphical optimisation focuses on identifying the peak (maximum) or valley (minimum) points on a graph, which represent the best or worst outcomes of a function. Options A, C, and D describe other aspects of graph analysis but not the core idea of optimisation.
Real World Connection
In the Real World
Think about your favourite food delivery app like Swiggy or Zomato. They use optimisation graphically (and mathematically) to find the shortest route for a delivery rider to reach you, saving time and fuel. This helps them deliver your order faster and more efficiently, making both customers and riders happy.
Key Vocabulary
Key Terms
MAXIMUM: The highest point or value a function reaches on a graph. | MINIMUM: The lowest point or value a function reaches on a graph. | OBJECTIVE FUNCTION: The equation or function we want to maximise or minimise. | OPTIMAL SOLUTION: The best possible outcome or value found through optimisation. | VERTEX: The turning point of a parabola, which is either its maximum or minimum.
What's Next
What to Learn Next
Next, you can explore 'Optimisation using Calculus'. This will teach you how to find these maximum and minimum points mathematically, without always needing to draw a graph, by using derivatives. It's a powerful tool that builds directly on understanding the graphical concept.
