Applications of Graphs
Graphs are an effective tool to represent relationships between two variables visually. They help in understanding patterns, trends, and variations in data. The following are some real-life applications of graphs.
1. Relationship Between Two Variables in Everyday Life
In daily life, we often observe that one quantity affects another. For example:
- If more electricity is consumed, the bill increases.
- If less electricity is used, the bill remains lower.
Here, the amount of electricity consumed is the independent variable, and the electricity bill is the dependent variable because the bill depends on the electricity consumed.
Think and Discuss
If you buy petrol for your car, the amount you pay depends on the litres of petrol you purchase.
- Which is the independent variable?
- Which is the dependent variable?
2. Example: Quantity and Cost of Petrol
Let’s consider the relationship between the quantity of petrol purchased and its cost. The data is given in the table below:
| Number of Litres of Petrol | 10 | 15 | 20 | 25 |
|---|---|---|---|---|
| Cost of Petrol (in ₹) | 500 | 750 | 1000 | 1250 |
Steps to Plot the Graph
- Choose a suitable scale for both axes.
- Mark the number of litres of petrol on the horizontal (x-axis).
- Mark the cost of petrol on the vertical (y-axis).
- Plot the points: (10,500), (15,750), (20,1000), (25,1250).
- Join the points to form a straight line.
Since this is a direct variation (cost increases proportionally with petrol quantity), the graph will be a linear graph passing through the origin (0,0).
Use of Graph for Estimation
- To find the cost of 12 litres of petrol: Locate 12 on the x-axis, draw a vertical line to the graph, then move horizontally to the y-axis to get the cost.
- To find how much petrol can be purchased for ₹ 800: Locate 800 on the y-axis, draw a horizontal line to the graph, then move vertically down to the x-axis to find the litres of petrol.
3. Example: Principal and Simple Interest
A bank offers 10% Simple Interest (S.I.) per year on deposits for senior citizens. The relation between deposit amount and simple interest earned is shown in the table below:
| Sum Deposited (in ₹) | 100 | 200 | 300 | 500 | 1000 |
|---|---|---|---|---|---|
| Annual S.I. (in ₹) | 10 | 20 | 30 | 50 | 100 |
Steps to Draw the Graph
- Choose a suitable scale:
- 1 unit = ₹100 on the x-axis
- 1 unit = ₹10 on the y-axis
- Mark Deposit amount along the horizontal (x-axis).
- Mark Simple Interest (S.I.) along the vertical (y-axis).
- Plot the points: (100,10), (200,20), (300,30), (500,50), (1000,100).
- Join the points to form a linear graph.
Use of Graph for Estimation
- To find interest on ₹250 deposit: Locate 250 on the x-axis, move vertically to the graph, then move horizontally to get ₹25 on the y-axis.
- To find deposit required for ₹70 interest: Locate 70 on the y-axis, move horizontally to the graph, then move vertically down to get ₹700 on the x-axis.
Is This a Case of Direct Variation?
Yes! The interest earned increases proportionally with the deposit, forming a linear relationship.
4. Example: Time and Distance Graph
Ajit rides a scooter at a constant speed of 30 km per hour. The time taken and distance covered are recorded in the table below:
| Time (in Hours) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Distance (in km) | 30 | 60 | 90 | 120 |
Steps to Plot the Graph
- Choose a scale:
- 2 units = 1 hour on x-axis
- 1 unit = 10 km on y-axis
- Mark time in hours on the horizontal (x-axis).
- Mark distance in km on the vertical (y-axis).
- Plot the points: (1,30), (2,60), (3,90), (4,120).
- Join the points to form a straight-line graph.
Use of Graph for Estimation
- To find time needed to cover 75 km: Locate 75 on the y-axis, move horizontally to the graph, then move vertically down to get 2.5 hours on the x-axis.
- To find distance covered in 3.5 hours: Locate 3.5 on the x-axis, move vertically to the graph, then move horizontally to get 105 km on the y-axis.
Since the speed is constant, the graph is linear.
Conclusion
- Graphs help visualize relationships between variables like cost vs. quantity, interest vs. deposit, and distance vs. time.
- Linear graphs represent direct proportionality, making it easy to estimate values.
- Real-life applications include electricity bills, fuel costs, bank interests, and travel distances.
Using these graphical methods, one can predict values, analyze trends, and make informed decisions efficiently. 🚀