The Modeling Process, Proportionality, and Geometric Similarity

Modeling Procedure

1. Observe real-world behavior and simplify factors
2. Conjecture relationships among factors and create rough model
3. Use mathmatical analysis to conclude the model
4. Interpret mathematical conclusions in terms of the real-world problem Conclusions pertain only to the model not real-world system because of somplifications. It is possible that the conclusion does not fit the real world system, then we have to refine the model or create a new model.

Mathematical Model

A mathematical model is representation of real-wolrd system or phenomeon.

Model Types

Experiments - obsevations are obtained directly Simulations - obsevations are obtained indirectly

Properties of a Model

Fidelity: The degree to which a model precisely represents real-world conditions. We expect greatest fidelity. However, models lose fidelity whenever real-world conditions are simplified.

Cost: The overall cost involved in developing and maintaining the model. We expect the models be the least expensive.

Flexibility: The capacity to modify the model based on newly collected data.

Construction of Models

STEP 1: Identify the problem Write problem restatement. STEP 2: Make assumptions Assumptions is helpful to simplify problems. You also need to justify your assumptions. First, you find all the factors and make assumption to simplify their relationships. Second, some factors (or variables) are neglectable. Neglect them will make the problem easier. STEP 3: Create the model Interpret what the model shows. Sometimes the model cannot interpret the real-world, this is when we need to go back to Step 1 and Step 2 redefine the problem. STEP 4: Test the model First, test if the model make common sense. Second, use actual data to test model. STEP 5: Use the model Operate model that users can understand. STEP 6: Maintain the model Adjust submodules if needed.

Similar to scientific method.

Iterative Nature of Model Construction

Repeate the model construction process to refine the model. This is opposite way to simplification.

Robust: A model is robust if its conclusions remain valid despite variations in assumptions. Fragile: A model is fragile if its conclusions depend heavily on precise conditions. Sensitivity: The degree of change in a model’s conclusions when some conditions change. The greater the conclusions change, the more sensitive the model is.

Modeling Using Proportionality

A rule for proportionality: $y\propto x$ and $x\propto y$, then $y\propto z$ In the graph, the straight line must pass through the origin, which means in formula $y=kx+b$ , b has to be zero. Or the proportionality is incorrect. We expect larger slope, k, and smaller initial displacement, b.

Modeling Using Geometric Similarity

Definition: Two or more objects have same shape but different size. This means corresponding angles are equal and corresponding distances are proportional.

If one object is geometric similar to another one, and the third object is geometric similar to the second one, it is always true that the first object is geometric similar to the third one. Different objects are scaled by some constant k. For example, if rectangle $ABCD$ is coresponding geometric similar to $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$, the relation ship is:

$$\frac{AB}{A^{\prime }{B}^{\prime }}=\frac{AC}{A^{\prime }{C}^{\prime }}=\frac{AD}{A^{\prime }{D}^{\prime }}=\frac{BC}{B^{\prime }{C}^{\prime }}=\frac{BD}{B^{\prime }{D}^{\prime }}=\frac{CD}{C^{\prime }{D}^{\prime }}=k$$

Similarly, the ratio of their surface areas are $S$ and $S^{\prime }$ with relationship:

Let’s say $AB=l$ and $A^{\prime }{B}^{\prime }=l^{\prime }$, so $\frac{AB}{A^{\prime }{B}^{\prime }}=\frac{l}{l^{\prime }}=k$ Combining the two equation, $$\frac{S}{S’} = k^2 = \frac{l^2}{l’^2}

Therefore, $$\frac{S}{l^2} = \frac{S'}{l'^2} = constant