Grade: Grade 12 Subject: Mathematics Unit: Real-World Modeling SAT: ProblemSolving+DataAnalysis ACT: Math

Mathematical Modeling

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What is Mathematical Modeling?

Mathematical modeling is the process of using mathematics to represent, analyze, and make predictions about real-world phenomena. It bridges the gap between abstract mathematical concepts and practical applications in science, engineering, economics, and everyday life.

Mathematical Model

A mathematical model is a description of a system using mathematical concepts and language. It translates real-world problems into mathematical form, allowing us to use mathematical tools to analyze and solve them.

The Modeling Process

Creating an effective mathematical model follows a systematic cycle:

The Modeling Cycle

  1. Identify the Problem: Define what you want to understand or predict
  2. Make Assumptions: Simplify the real-world situation by identifying key factors
  3. Build the Model: Translate assumptions into mathematical equations
  4. Solve and Analyze: Use mathematical techniques to find solutions
  5. Validate: Compare model predictions with real data
  6. Refine: Adjust the model based on validation results

Types of Mathematical Models

1. Linear Models

y = mx + b

  • Used when there is a constant rate of change
  • Examples: Cost functions, simple depreciation, constant velocity
  • Easy to work with but limited in scope

2. Exponential Models

y = a * b^t or y = a * e^(kt)

  • Growth (k > 0): Population growth, compound interest, viral spread
  • Decay (k < 0): Radioactive decay, cooling, depreciation
  • Rate of change is proportional to current value

3. Polynomial Models

y = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0

  • Quadratic: Projectile motion, area optimization
  • Cubic and higher: Complex curves, interpolation
  • Flexible but can behave unpredictably outside the data range

4. Logistic Models

P(t) = L / (1 + e^(-k(t - t_0)))

  • Models growth with a carrying capacity (upper limit)
  • Examples: Population with limited resources, disease spread, market saturation
  • L = carrying capacity, k = growth rate

Choosing the Right Model

Pattern in DataSuggested ModelKey Indicator
Constant increase/decreaseLinearConstant first differences
Accelerating increase/decreaseExponentialConstant ratio between consecutive values
Single maximum or minimumQuadraticConstant second differences
S-shaped curveLogisticGrowth slows as it approaches a limit
Repeating patternTrigonometricPeriodic oscillations

Model Parameters and Fitting

Once you choose a model type, you need to determine the specific parameter values that best fit your data.

Regression Analysis

Regression finds the parameter values that minimize the difference between predicted and actual values. The most common method is least squares regression, which minimizes the sum of squared residuals.

Residual = Actual Value - Predicted Value

A good fit has small residuals that appear randomly scattered (no pattern).

Interpreting Model Parameters

Each parameter in a model has real-world meaning:

  • Linear (y = mx + b): m = rate of change per unit, b = initial value
  • Exponential (y = a*b^t): a = initial value, b = growth/decay factor per time unit
  • Quadratic (y = ax^2 + bx + c): a determines concavity and width, vertex gives optimal value

Examples

Example 1: Population Growth Model

Problem: A city had a population of 50,000 in 2000 and 65,000 in 2010. Assuming exponential growth, predict the population in 2025.

Solution:

Use the exponential model: P(t) = P_0 * e^(kt)

Step 1: Set up with known values (let t = 0 be year 2000)

P_0 = 50,000 and P(10) = 65,000

Step 2: Find k

65,000 = 50,000 * e^(10k)

1.3 = e^(10k)

ln(1.3) = 10k

k = ln(1.3)/10 = 0.0262

Step 3: Predict for 2025 (t = 25)

P(25) = 50,000 * e^(0.0262 * 25)

P(25) = 50,000 * e^(0.655)

P(25) = 50,000 * 1.925 = 96,250

Example 2: Choosing the Right Model

Problem: Data for a new product's sales over 6 months:

Month123456
Sales10018032458310501890

Determine the best model type.

Solution:

Check for linear: First differences: 80, 144, 259, 467, 840 (not constant - not linear)

Check for exponential: Ratios: 180/100=1.8, 324/180=1.8, 583/324=1.8, 1050/583=1.8, 1890/1050=1.8

The ratios are constant (approximately 1.8), so this is exponential growth.

Model: S(t) = 100 * (1.8)^(t-1) or equivalently S(t) = 55.6 * (1.8)^t

Example 3: Quadratic Optimization

Problem: A farmer has 200 meters of fencing to enclose a rectangular field along a river (no fence needed along the river). What dimensions maximize the area?

Solution:

Step 1: Define variables

Let x = width perpendicular to river, y = length along river

Step 2: Set up constraint

2x + y = 200 (only three sides need fencing)

So y = 200 - 2x

Step 3: Create area function

A(x) = x * y = x(200 - 2x) = 200x - 2x^2

Step 4: Find maximum (vertex of parabola)

x = -b/(2a) = -200/(2*(-2)) = 50 meters

y = 200 - 2(50) = 100 meters

Maximum area: A = 50 * 100 = 5,000 square meters

Example 4: Logistic Growth

Problem: A disease spreads in a community of 10,000 people. Initially 100 are infected, and after 5 days, 1,000 are infected. Model the spread and predict when 5,000 will be infected.

Solution:

Use logistic model: P(t) = L / (1 + ((L-P_0)/P_0) * e^(-kt))

L = 10,000 (carrying capacity), P_0 = 100 (initial infected)

Step 1: Find k using P(5) = 1000

1000 = 10000 / (1 + 99*e^(-5k))

1 + 99*e^(-5k) = 10

e^(-5k) = 9/99 = 0.0909

k = -ln(0.0909)/5 = 0.479

Step 2: Find when P(t) = 5000

5000 = 10000 / (1 + 99*e^(-0.479t))

1 + 99*e^(-0.479t) = 2

e^(-0.479t) = 1/99

t = ln(99)/0.479 = 9.6 days

Example 5: Model Validation

Problem: A linear model y = 2.5x + 10 predicts product demand. Actual data shows: x=5, y=22; x=10, y=36; x=15, y=47. Evaluate the model.

Solution:

Calculate predictions and residuals:

xActualPredictedResidualResidual^2
52222.5-0.50.25
10363511
154747.5-0.50.25

Sum of squared residuals: 0.25 + 1 + 0.25 = 1.5

Evaluation: Residuals are small and don't show a pattern, suggesting the linear model is appropriate. The model slightly underpredicts at x=10 but is otherwise accurate.

Practice

Apply your understanding of mathematical modeling to solve these problems.

1. A bacteria culture starts with 500 bacteria and doubles every 3 hours. Which model type is most appropriate?

A) Linear   B) Quadratic   C) Exponential   D) Logistic

2. Using the bacteria from problem 1, how many bacteria will there be after 12 hours?

A) 2,000   B) 4,000   C) 8,000   D) 16,000

3. A ball is thrown upward with height h(t) = -16t^2 + 64t + 5 feet. What is the maximum height?

A) 64 ft   B) 69 ft   C) 85 ft   D) 133 ft

4. Data points: (1, 8), (2, 11), (3, 14), (4, 17). What type of model fits best?

A) Linear   B) Quadratic   C) Exponential   D) Logistic

5. For the linear model y = 3x + 5, what does the slope represent?

A) Initial value   B) Rate of change   C) Maximum value   D) Minimum value

6. A population is modeled by P(t) = 500 * (1.05)^t. What is the annual growth rate?

A) 0.5%   B) 5%   C) 50%   D) 105%

7. Which situation would best be modeled by a logistic function?

A) Free-falling object   B) Compound interest   C) Spread of a rumor in a school   D) Distance vs time at constant speed

8. A car depreciates 15% per year from an initial value of $30,000. What is the value after 3 years?

A) $16,500   B) $18,403   C) $21,675   D) $25,500

9. A residual plot shows a clear curved pattern. This suggests:

A) The model is perfect   B) A non-linear model might be better   C) More data is needed   D) The y-intercept is wrong

10. A company's profit P(x) = -2x^2 + 100x - 800 depends on price x. What price maximizes profit?

A) $20   B) $25   C) $50   D) $100

Click to reveal answers
  1. C) Exponential - doubling indicates constant ratio
  2. C) 8,000 - After 12 hours (4 doubling periods): 500 * 2^4 = 8,000
  3. B) 69 ft - Vertex at t = 2: h(2) = -16(4) + 64(2) + 5 = 69
  4. A) Linear - First differences are constant (3)
  5. B) Rate of change - slope represents change in y per unit change in x
  6. B) 5% - Growth factor 1.05 means 5% growth rate
  7. C) Spread of a rumor in a school - limited population creates carrying capacity
  8. B) $18,403 - 30000 * (0.85)^3 = 18,403.125
  9. B) A non-linear model might be better - patterns in residuals indicate model inadequacy
  10. B) $25 - Vertex at x = -100/(2*(-2)) = 25

Check Your Understanding

1. Explain the difference between exponential growth and logistic growth. When would each be appropriate?

Show answer

Exponential growth assumes unlimited resources and grows without bound - appropriate for early-stage population growth, compound interest, or bacterial growth in abundant conditions. Logistic growth includes a carrying capacity that limits growth - appropriate when resources are limited, like population in a confined environment or market saturation for a product.

2. Why is model validation important? What could happen if you skip this step?

Show answer

Model validation compares predictions with real data to ensure the model accurately represents reality. Skipping this step could lead to using a model that makes poor predictions, resulting in bad decisions. A model might fit training data well but fail to generalize. Validation helps identify when a model needs refinement or when a different model type is needed.

3. What assumptions are typically made when using a linear model, and when might these assumptions be violated?

Show answer

Linear models assume: constant rate of change, no upper/lower bounds, and additive effects. These assumptions are violated when: growth rate depends on current value (exponential), there are natural limits (logistic), relationships curve (polynomial), or changes aren't constant. For example, salary growth often isn't linear - it may accelerate with experience.

4. How do you interpret the parameters in an exponential decay model like A(t) = A_0 * e^(-kt)?

Show answer

A_0 is the initial amount at time t=0. The parameter k is the decay constant - larger k means faster decay. The half-life (time for half to decay) is ln(2)/k. The ratio e^(-k) tells you the fraction remaining after one time unit. For example, if k=0.1 per year, about 90.5% remains after each year, and half-life is about 6.9 years.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review