Grade: Grade 9 Subject: Mathematics Unit: Functions & Modeling SAT: ProblemSolving+DataAnalysis ACT: Math

Modeling Real Situations

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Mathematical Modeling

Mathematical modeling is the process of using mathematical expressions, equations, or functions to represent real-world situations. A good model captures the essential features of a situation and allows us to make predictions or analyze behavior.

When we model real situations with functions, we translate words and relationships into mathematical language. This allows us to use algebra to answer questions about the real world.

Steps for Creating a Mathematical Model

  1. Identify variables: What quantities change? Which is independent (input) and which is dependent (output)?
  2. Identify relationships: How are the variables related? Is it linear, quadratic, exponential?
  3. Write the function: Express the relationship using function notation
  4. Define the domain: What input values make sense in context?
  5. Validate: Does the model give reasonable outputs for known inputs?

Common Types of Models

Model Type Function Form Real-World Examples
Linear f(x) = mx + b Cost = rate × quantity + fixed fee; Distance = speed × time
Quadratic f(x) = ax² + bx + c Projectile motion; Area problems
Exponential Growth f(x) = a · bˣ (b > 1) Population growth; Compound interest
Exponential Decay f(x) = a · bˣ (0 < b < 1) Radioactive decay; Depreciation

Key Vocabulary in Word Problems

Phrase Mathematical Meaning
"initial value," "starting amount," "flat fee" The y-intercept (b in y = mx + b)
"rate," "per," "each" The slope or rate of change (m)
"total," "combined," "altogether" The output of the function
"doubles," "triples," "halves" Exponential model indicator

SAT/ACT Connection

Real-world modeling problems are heavily tested on both the SAT and ACT. The SAT's "Problem Solving and Data Analysis" domain specifically tests your ability to create and interpret models. Look for key words that indicate the type of function needed.

Examples

Work through these examples to see how to build and use mathematical models.

Example 1: Linear Cost Model

Problem: A streaming service charges $12 per month plus a one-time activation fee of $25. Write a function C(m) for the total cost after m months, then find the cost for 1 year.

Step 1: Identify the components

Fixed cost (one-time): $25

Rate per month: $12

Step 2: Write the function

C(m) = 12m + 25

Step 3: Calculate for 1 year (12 months)

C(12) = 12(12) + 25 = 144 + 25 = 169

Answer: The cost for 1 year is $169.

Example 2: Distance-Rate-Time Model

Problem: A car travels at a constant speed of 55 mph. Write a function D(t) for distance traveled after t hours. How long to travel 220 miles?

Step 1: Use distance = rate × time

D(t) = 55t

Step 2: To find time for 220 miles, solve D(t) = 220

55t = 220

t = 4

Answer: It takes 4 hours to travel 220 miles.

Example 3: Depreciation Model

Problem: A computer worth $1,200 loses $150 in value each year. Write V(t) for its value after t years. When is it worth $600?

Step 1: Identify starting value and rate of decrease

Initial value: $1,200; Decrease: $150 per year

Step 2: Write the function (linear decay)

V(t) = 1200 - 150t

Step 3: Solve V(t) = 600

1200 - 150t = 600

-150t = -600

t = 4

Answer: The computer is worth $600 after 4 years.

Example 4: Area Model

Problem: A rectangular garden has length 3 feet more than its width w. Write A(w) for the area. Find the area if the width is 8 feet.

Step 1: Express length in terms of width

Length = w + 3

Step 2: Write the area function

A(w) = w(w + 3) = w² + 3w

Step 3: Evaluate at w = 8

A(8) = 8² + 3(8) = 64 + 24 = 88

Answer: The area is 88 square feet.

Example 5: Interpreting a Model

Problem: The function P(t) = 500(1.03)ᵗ models a population after t years. What does 500 represent? What does 1.03 represent?

Interpreting 500:

When t = 0, P(0) = 500(1.03)⁰ = 500(1) = 500

So 500 is the initial population.

Interpreting 1.03:

The base 1.03 = 1 + 0.03, indicating 3% growth per year.

Each year, the population is multiplied by 1.03 (increases by 3%).

Answer: 500 is the initial population; 1.03 indicates 3% annual growth rate.

Practice

Create and use models to answer these questions.

1. A taxi charges $3.50 plus $2.25 per mile. What function models the cost C for m miles?

A) C(m) = 3.50m + 2.25   B) C(m) = 2.25m + 3.50   C) C(m) = 5.75m   D) C(m) = 2.25m - 3.50

2. Using C(m) = 2.25m + 3.50, find the cost for a 10-mile trip.

A) $22.50   B) $26.00   C) $57.50   D) $35.00

3. A pool is draining at 25 gallons per minute. It starts with 1,500 gallons. What function models water remaining after t minutes?

A) W(t) = 25t + 1500   B) W(t) = 1500 - 25t   C) W(t) = 1500t - 25   D) W(t) = 1500 + 25t

4. Using W(t) = 1500 - 25t, when is the pool empty?

A) 25 minutes   B) 60 minutes   C) 75 minutes   D) 100 minutes

5. A bacteria population doubles every hour starting with 100. Which function models this?

A) P(t) = 100 + 2t   B) P(t) = 100(2)ᵗ   C) P(t) = 2(100)ᵗ   D) P(t) = 200t

6. In P(t) = 2000(0.85)ᵗ, what does 0.85 tell us?

A) 85% growth per period   B) 15% decay per period   C) Initial value of 85   D) 85 periods

7. A rectangle has length 2w - 1 and width w. What is A(w)?

A) A(w) = 2w² - 1   B) A(w) = 2w² - w   C) A(w) = 3w - 1   D) A(w) = 2w - w²

8. The function h(t) = -16t² + 48t + 4 models height in feet after t seconds. What is the initial height?

A) -16 feet   B) 48 feet   C) 4 feet   D) 0 feet

9. A phone plan costs $40/month plus $5 per GB of data. If the bill is $65, how many GB were used?

A) 3 GB   B) 5 GB   C) 8 GB   D) 13 GB

10. A car's value V(t) = 20000(0.90)ᵗ. What is the value after 2 years?

A) $16,200   B) $18,000   C) $14,580   D) $36,000

Click to reveal answers
  1. B) C(m) = 2.25m + 3.50 - $2.25 per mile (slope) plus $3.50 base (y-intercept)
  2. B) $26.00 - C(10) = 2.25(10) + 3.50 = 22.50 + 3.50 = 26.00
  3. B) W(t) = 1500 - 25t - Starts at 1500, decreases by 25 each minute
  4. B) 60 minutes - 1500 - 25t = 0; t = 60
  5. B) P(t) = 100(2)ᵗ - Exponential doubling with initial value 100
  6. B) 15% decay per period - 0.85 = 1 - 0.15, so 15% is lost each period
  7. B) A(w) = 2w² - w - Area = w(2w - 1) = 2w² - w
  8. C) 4 feet - At t = 0, h(0) = 4; initial height is the constant term
  9. B) 5 GB - 40 + 5g = 65; 5g = 25; g = 5
  10. A) $16,200 - V(2) = 20000(0.90)² = 20000(0.81) = 16,200

Check Your Understanding

Answer these reflection questions to deepen your understanding.

1. How do you determine whether a real-world situation should be modeled with a linear or exponential function?

Reveal Answer

Linear models are appropriate when the rate of change is constant - the quantity increases or decreases by the same amount each time period (e.g., "$5 per hour"). Exponential models are appropriate when the rate of change is proportional to the current amount - the quantity increases or decreases by the same percentage each time period (e.g., "doubles every year" or "loses 10% each month"). Look for words like "per" (linear) vs. "doubles/halves/percent" (exponential).

2. In the function C(x) = 25x + 100, what do the 25 and 100 represent in real-world terms?

Reveal Answer

The 25 represents the rate of change or unit cost - the cost increases by $25 for each additional unit of x. The 100 represents the initial or fixed cost - this is charged regardless of how many units are purchased. For example, if this models a service, $100 might be a setup fee and $25 might be the cost per hour of service.

3. Why is it important to consider the domain when creating a real-world model?

Reveal Answer

The domain must reflect what makes sense in the real-world context. For example, if modeling time, negative values usually don't make sense. If modeling the number of items, we need whole numbers. If modeling a pool draining, the domain is limited to when water remains (before the pool is empty). Without considering domain, the model might give mathematically valid but meaningless answers (like negative people or fractional cars).

4. What does it mean to "interpret" a mathematical model in context?

Reveal Answer

Interpreting a model means explaining what the mathematical components mean in real-world terms. For example, in h(t) = -16t² + 64t + 5, we interpret: the 5 as the initial height (feet), the 64 relates to initial velocity (ft/sec), and the -16 comes from gravity. We also interpret outputs: "h(2) = 69 means the object is 69 feet high after 2 seconds." Interpretation connects abstract math to concrete meaning.

🚀 Next Steps

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