MODELING

Modeling

Definition Of Modeling

Modeling is the process of representing real life situations through equations or inequalities.

Examples of Modeling

Suppose a laboratory prepares a 200-mg of technetium-99m which has a half-life of 6 hours. Its exponential decay can be modeled by the equation y = 200(0.5)^x

Video Examples: Mathematical Modeling

Solved Example on Modeling

Ques: The population of the world(in million) is modeled by the expression 0.75x² + 74.5x + 4600, where x represents the number of years since 1985. Estimate the world population in 1995.

Choices:

A. 4,420 millions
B. 7,420 millions
C. 6,420 millions
D. 5,420 millions
Correct Answer: D

Solution:

Step 1: [Original model.] 0.75x² + 74.5x + 4600
Step 2: [Replace x = 10 in the model.] 0.75(10)² + 74.5(10) + 4600
Step 3: [Simplify.] = (0.75 × 100) + (74.5 × 10) + 4600
Step 4: = 75 + 745 + 4600
Step 5: = 5420
Step 6: So, the world population in 1995 was 5,420 millions.

Real-world Connections

Mathematical models are used in sciences such as biology and physics and in other fields such as economics and sociology

Quick Summary

Mathematical models use equations to represent real-world scenarios.
Models can be used to predict future outcomes based on current data.
Exponential decay is a common type of mathematical model.

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🍎 Teacher Insights

Emphasize the importance of understanding the context of the problem and correctly identifying variables. Encourage students to practice applying models to real-world situations.

🎓 Prerequisites

Algebra
Basic equation solving
Understanding of variables and functions

Check Your Knowledge

Q1: A population grows according to the model P(t) = 1000 * 2^t, where t is time in years. What is the population after 3 years?

A. 2000 B. 3000 C. 8000 D. 16000

Q2: Which of the following is an example of mathematical modeling?

A. Solving a quadratic equation B. Using a formula to calculate area C. Representing the spread of a disease with differential equations D. Graphing a linear function

Frequently Asked Questions

Q: What are the real-world applications of mathematical modeling?
A: Mathematical models are used in various fields such as biology, physics, economics, and sociology to understand and predict phenomena.

Q: How do I choose the right equation for a given scenario?
A: Analyze the scenario carefully to determine the relationships between the variables and choose an equation that accurately represents those relationships.