Grade: Grade 11 Subject: Mathematics Unit: Functions & Modeling SAT: ProblemSolving+DataAnalysis ACT: Math

Real-World Modeling

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Mathematical Modeling

Mathematical modeling is the process of using mathematical concepts and language to describe, analyze, and predict real-world phenomena. A mathematical model is an equation, function, or system of equations that represents a relationship observed in data or described in a problem.

The Modeling Process

Creating an effective mathematical model follows a systematic process:

  1. Identify the Problem: Clearly define what you want to model or predict
  2. Gather Data: Collect relevant information and measurements
  3. Choose a Model Type: Select an appropriate function type based on the data pattern
  4. Build the Model: Determine the specific equation using data points or regression
  5. Test and Validate: Check if the model accurately predicts known values
  6. Apply and Interpret: Use the model to make predictions and explain results in context

Choosing the Right Model

Different real-world situations call for different function types:

Model Type Form Best Used When Real-World Examples
Linear y = mx + b Constant rate of change Distance traveled at constant speed, simple depreciation, hourly wages
Quadratic y = ax² + bx + c Parabolic path, max/min optimization Projectile motion, profit optimization, area problems
Exponential y = a · bˣ Constant percent change, multiplicative growth/decay Population growth, radioactive decay, compound interest
Logarithmic y = a · ln(x) + b Rapid initial change that slows over time Sound intensity, earthquake magnitude, learning curves
Rational y = a/(x - h) + k Inverse relationships with limits Speed vs. time, concentration dilution, resistance
Sinusoidal y = a·sin(bx) + c Periodic, repeating behavior Tides, temperature cycles, sound waves

Building Models from Data

Regression

Regression is a statistical method for finding the best-fit equation that describes the relationship between variables. Your calculator or software can perform linear, quadratic, exponential, and other types of regression.

When given a table of data:

  • Plot the points to visualize the pattern
  • Look for linear (straight), curved (quadratic/exponential), or periodic patterns
  • Check if differences (linear) or ratios (exponential) are constant
  • Use regression to find the equation, then verify with known points

Key Formulas for Common Models

Linear Model (Two Points)

m = (y₂ - y₁)/(x₂ - x₁)

y - y₁ = m(x - x₁)

Exponential Model (Two Points)

Given (x₁, y₁) and (x₂, y₂):

b = (y₂/y₁)^(1/(x₂-x₁))

a = y₁/b^(x₁)

Quadratic Model (Vertex Form)

Given vertex (h, k) and another point (x, y):

a = (y - k)/(x - h)²

f(x) = a(x - h)² + k

Domain Restrictions and Reasonableness

Real-world models have practical limitations:

  • Domain restrictions: Time can't be negative; prices must be positive
  • Range restrictions: Population can't be negative; percentages cap at 100%
  • Model validity: Models may only be accurate within a certain range of the data used to create them
  • Extrapolation caution: Predictions far beyond the data range become unreliable

💡 Examples

Example 1: Choosing a Model Type

Problem: The following data shows the number of bacteria in a culture over time:

Time (hours)01234
Bacteria100150225338507

Determine the appropriate model type and write an equation.

Step 1: Check for constant differences (linear test):

150 - 100 = 50, 225 - 150 = 75, 338 - 225 = 113, 507 - 338 = 169

Differences are NOT constant, so not linear.

Step 2: Check for constant ratios (exponential test):

150/100 = 1.5, 225/150 = 1.5, 338/225 ≈ 1.5, 507/338 ≈ 1.5

Ratios are approximately constant at 1.5!

Step 3: Write the exponential model:

Initial value a = 100, growth factor b = 1.5

P(t) = 100(1.5)ᵗ

Step 4: Verify: P(2) = 100(1.5)² = 100(2.25) = 225 ✓

Answer: P(t) = 100(1.5)ᵗ where t is time in hours

Example 2: Projectile Motion (Quadratic Model)

Problem: A ball is thrown upward from a height of 6 feet with an initial velocity of 40 ft/s. The height h after t seconds is given by h(t) = -16t² + 40t + 6. Find the maximum height and when the ball hits the ground.

Finding Maximum Height:

The maximum occurs at the vertex. For h(t) = -16t² + 40t + 6:

t = -b/(2a) = -40/(2·(-16)) = -40/(-32) = 1.25 seconds

h(1.25) = -16(1.25)² + 40(1.25) + 6

h(1.25) = -16(1.5625) + 50 + 6 = -25 + 56 = 31 feet

Finding When Ball Hits Ground:

Set h(t) = 0: -16t² + 40t + 6 = 0

Using quadratic formula: t = (-40 ± √(1600 + 384))/(-32)

t = (-40 ± √1984)/(-32) = (-40 ± 44.54)/(-32)

t = (-40 - 44.54)/(-32) = 2.64 seconds (positive solution)

Answer: Maximum height is 31 feet at 1.25 seconds; ball hits ground at ≈ 2.64 seconds

Example 3: Building a Model from Two Points

Problem: A car depreciates in value. After 2 years it's worth $18,000, and after 5 years it's worth $12,000. Assuming linear depreciation, write a model and find the original price.

Step 1: Identify the points: (2, 18000) and (5, 12000)

Step 2: Find the slope (rate of depreciation):

m = (12000 - 18000)/(5 - 2) = -6000/3 = -2000 per year

Step 3: Write the equation using point-slope form:

V - 18000 = -2000(t - 2)

V = -2000t + 4000 + 18000

V(t) = -2000t + 22000

Step 4: Find original price (t = 0):

V(0) = -2000(0) + 22000 = $22,000

Answer: V(t) = -2000t + 22000; original price was $22,000

Example 4: Compound Interest Model

Problem: You invest $5,000 at 4.5% annual interest compounded monthly. Write a model for the balance after t years, and find how long until the investment doubles.

Step 1: Use the compound interest formula:

A(t) = P(1 + r/n)^(nt)

Where P = 5000, r = 0.045, n = 12 (monthly)

Step 2: Write the model:

A(t) = 5000(1 + 0.045/12)^(12t)

A(t) = 5000(1.00375)^(12t)

Step 3: Find doubling time (A = 10000):

10000 = 5000(1.00375)^(12t)

2 = (1.00375)^(12t)

ln(2) = 12t · ln(1.00375)

t = ln(2)/(12 · ln(1.00375))

t = 0.693/(12 · 0.00374) = 0.693/0.0449 ≈ 15.4 years

Answer: A(t) = 5000(1.00375)^(12t); doubles in approximately 15.4 years

Example 5: Interpreting and Using a Model

Problem: The population of a town is modeled by P(t) = 25000 · e^(0.018t), where t is years since 2010. (a) What was the population in 2010? (b) What is the growth rate? (c) When will the population reach 35,000?

(a) Population in 2010 (t = 0):

P(0) = 25000 · e^0 = 25000 · 1 = 25,000 people

(b) Growth rate:

The model P = P₀e^(rt) has r = 0.018

Growth rate = 1.8% per year

(c) When P = 35,000:

35000 = 25000 · e^(0.018t)

1.4 = e^(0.018t)

ln(1.4) = 0.018t

t = ln(1.4)/0.018 = 0.336/0.018 ≈ 18.7 years

Year: 2010 + 18.7 ≈ late 2028

Answer: (a) 25,000 (b) 1.8% per year (c) Late 2028

✏️ Practice

Solve these problems to strengthen your modeling skills.

Problem 1: A hot cup of coffee cools from 180°F to 120°F in 10 minutes while in a 70°F room. The data suggests exponential decay toward room temperature. If the temperature after t minutes is modeled by T(t) = 70 + 110e^(-kt), find k.

Problem 2: A company's profit is modeled by P(x) = -2x² + 120x - 400, where x is the number of items sold (in hundreds). Find the number of items that maximizes profit and the maximum profit.

Problem 3: The following data shows website visitors over months after launch:

Month1234
Visitors2004008001600

Write an exponential model and predict visitors in month 6.

Problem 4: A population of 500 endangered animals grows at 3% per year. Write a model and find when the population will reach 1000.

Problem 5: A drug's concentration in the bloodstream decreases by 15% each hour. If the initial concentration is 200 mg/L, write a model and find the concentration after 6 hours.

Problem 6: A ball is dropped from 100 feet. Each bounce reaches 75% of the previous height. Write a model for the height after n bounces and find when the bounce height first drops below 10 feet.

Problem 7: The value of a house increases from $200,000 to $230,000 over 4 years. Assuming linear appreciation, write a model and predict the value after 10 years.

Problem 8: The number of bacteria doubles every 3 hours. Starting with 500 bacteria, write a model using base 2 and find the population after 12 hours.

Problem 9: A radioactive substance has a half-life of 8 days. If you start with 200 grams, write a model and find how much remains after 20 days.

Problem 10: The height of a firework is given by h(t) = -16t² + 128t + 5. Find the time when the firework reaches its maximum height and how long until it hits the ground.

Click to reveal answers
  1. k ≈ 0.0693 (Solve: 120 = 70 + 110e^(-10k) → 50/110 = e^(-10k) → k = -ln(5/11)/10)
  2. Maximum profit of $1400 when selling 3000 items (x = 30, vertex at x = -120/(2·(-2)) = 30)
  3. V(t) = 100(2)ᵗ; In month 6: V(6) = 100(2)⁶ = 6400 visitors
  4. P(t) = 500(1.03)ᵗ; Reaches 1000 when t = ln(2)/ln(1.03) ≈ 23.4 years
  5. C(t) = 200(0.85)ᵗ; C(6) = 200(0.85)⁶ ≈ 75.4 mg/L
  6. H(n) = 100(0.75)ⁿ; Below 10 feet when n > ln(0.1)/ln(0.75) ≈ 8 bounces (first at bounce 9)
  7. V(t) = 7500t + 200000; V(10) = $275,000
  8. P(t) = 500 · 2^(t/3); P(12) = 500 · 2⁴ = 8000 bacteria
  9. A(t) = 200(1/2)^(t/8); A(20) = 200(0.5)^2.5 ≈ 35.4 grams
  10. Maximum at t = 4 seconds (h = 261 feet); hits ground at t ≈ 8.04 seconds

✅ Check Your Understanding

Question 1: How do you determine whether data is better modeled by a linear function versus an exponential function?

Show answer

For linear data, the differences between consecutive y-values are constant. For exponential data, the ratios between consecutive y-values are constant. You can also plot the data: linear data forms a straight line, while exponential data forms a curve that either increases rapidly or decreases toward zero.

Question 2: What does the value of the base b tell you about an exponential model y = a · bˣ?

Show answer

If b > 1, the model represents exponential growth. If 0 < b < 1, it represents exponential decay. The value (b - 1) × 100% gives the percent change per unit of x. For example, b = 1.05 means 5% growth per unit, while b = 0.85 means 15% decay per unit.

Question 3: Why is it important to consider domain restrictions when creating real-world models?

Show answer

Real-world quantities have natural limitations that mathematical equations don't automatically capture. Time is usually non-negative, populations can't be negative or fractional, prices must be positive, and percentages can't exceed 100%. Without appropriate domain restrictions, a model might produce mathematically valid but physically meaningless results.

Question 4: In a quadratic model of projectile motion h(t) = at² + bt + c, what do the coefficients a, b, and c represent physically?

Show answer

The coefficient a relates to acceleration due to gravity (a = -16 ft/s² or -4.9 m/s² when measuring in standard units—negative because gravity pulls downward). The coefficient b represents the initial vertical velocity. The coefficient c represents the initial height from which the object is launched or dropped.

🚀 Next Steps

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