Solved Exercises on Mathematical Modeling in Grade 8

Master mathematical modeling: real-world applications, functions, and problem-solving through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Linear Function Model
Exercise 1
A car rental company charges a flat fee of $30 plus $0.25 per mile driven. Write a function to model the total cost and calculate the cost for driving 120 miles.
Definition:

Mathematical Model: A mathematical representation of a real-world situation using equations, functions, or formulas

Mathematical modeling process:
  1. Identify the variables and constants in the problem
  2. Translate the real-world situation into mathematical language
  3. Create the function or equation
  4. Use the model to solve specific problems
Variables
Cost, Miles
Function
C(m) = 30 + 0.25m
Solution
C(120) = $60
Step 1: Identify variables and constants

Variables: Total cost (C), miles driven (m)

Constants: Flat fee = $30, Rate = $0.25 per mile

Step 2: Translate to mathematical language

Total cost = Flat fee + (Rate per mile × Miles driven)

C = 30 + 0.25m

Step 3: Create the function

C(m) = 30 + 0.25m

Where C(m) represents cost as a function of miles

Step 4: Solve for specific case

For 120 miles: C(120) = 30 + 0.25(120) = 30 + 30 = $60

C(m) = 30 + 0.25m, C(120) = $60
Final answer:

The mathematical model is C(m) = 30 + 0.25m, and the cost for driving 120 miles is $60

Applied rules:

• Linear function: f(x) = mx + b, where m is rate and b is initial value

• Model validation: Check that the model makes sense in context

• Unit consistency: Ensure all units are compatible

2 Area Model
Exercise 2
A rectangular garden has a length that is 3 feet longer than its width. Write a function to model the area of the garden in terms of its width. If the width is 8 feet, what is the area?
Definition:

Area Model: A mathematical model that represents geometric relationships using area formulas

Width
w
Length
w + 3
Area Function
A(w) = w(w+3)
Step 1: Define the variable

Let w = width of the garden (in feet)

Step 2: Express length in terms of width

Length = width + 3 = w + 3

Step 3: Write the area formula

Area = length × width

A(w) = (w + 3) × w = w(w + 3) = w² + 3w

Step 4: Calculate area for specific width

When w = 8: A(8) = 8² + 3(8) = 64 + 24 = 88 square feet

A(w) = w² + 3w, A(8) = 88 sq ft
Final answer:

The area model is A(w) = w² + 3w, and when the width is 8 feet, the area is 88 square feet

Applied rules:

• Area formula: Rectangle area = length × width

• Quadratic function: Area model often results in quadratic functions

• Dimension consistency: All dimensions must use the same units

3 Population Growth Model
Exercise 3
A town's population increases by 2% each year. If the current population is 10,000, write a function to model the population after t years and find the population after 5 years.
Definition:

Exponential Growth Model: A mathematical model where a quantity increases by a fixed percentage over equal time periods

Initial Value
10,000
Growth Rate
1.02
Function
P(t) = 10,000(1.02)^t
Step 1: Identify the growth pattern

Population increases by 2% each year

Multiplier = 1 + 0.02 = 1.02

Step 2: Write the exponential function

P(t) = Initial population × (growth multiplier)^time

P(t) = 10,000 × (1.02)^t

Step 3: Calculate population after 5 years

P(5) = 10,000 × (1.02)^5

P(5) = 10,000 × 1.10408... ≈ 11,041 people

Step 4: Interpret the result

After 5 years, the population grows from 10,000 to approximately 11,041 people

P(t) = 10,000(1.02)^t, P(5) ≈ 11,041
Final answer:

The population model is P(t) = 10,000(1.02)^t, and after 5 years the population will be approximately 11,041 people

Applied rules:

• Exponential growth: P(t) = P₀(1 + r)^t

• Growth rate: Convert percentage to decimal (2% = 0.02)

• Multiplier: 1 + growth rate

Mathematical Modeling Fundamentals
f(x) = mx + b
Linear Function
Linear
f(x) = mx + b
Constant rate of change
Quadratic
f(x) = ax² + bx + c
Parabolic relationship
Exponential
f(x) = ab^x
Percent change
Key definitions:

Mathematical Model: A mathematical representation of a real-world situation

Function: A relationship where each input has exactly one output

Variable: A symbol that represents a changing quantity

Parameter: A constant value that defines the model

Domain: The set of possible input values

Range: The set of possible output values

Model Validation: Checking that the model accurately represents the situation

Linear Model: A model with constant rate of change

Mathematical Modeling Process:
  1. Problem identification: Understand what is being modeled
  2. Variable selection: Choose appropriate variables
  3. Relationship identification: Determine how variables relate
  4. Model creation: Write the mathematical equation
  5. Model validation: Check if model makes sense
  6. Model application: Use model to solve problems
Tip 1: Always define your variables clearly with units.
Tip 2: Check that your model makes sense in the real world.
Tip 3: Consider the domain restrictions based on the context.
Tip 4: Use the model to make predictions and test against reality.
Common errors: Forgetting units, misidentifying relationships, not checking reasonableness.
Exam preparation: Practice identifying variable relationships, creating models, and solving real-world problems.
Solution: Exercises 4 to 5
4 Physics Model
Exercise 4
The distance traveled by a freely falling object is given by d = 4.9t², where d is distance in meters and t is time in seconds. How far does an object fall in 3 seconds? How long does it take to fall 44.1 meters?
Definition:

Physics Model: A mathematical model that describes physical phenomena using scientific laws

Distance after 3s
d = 4.9(3)² = 44.1m
Time for 44.1m
t = √(44.1/4.9) = 3s
Step 1: Calculate distance after 3 seconds

Substitute t = 3 into d = 4.9t²

d = 4.9(3)² = 4.9(9) = 44.1 meters

Step 2: Find time to fall 44.1 meters

Substitute d = 44.1 into d = 4.9t²

44.1 = 4.9t²

t² = 44.1/4.9 = 9

t = √9 = 3 seconds

Step 3: Verify the solution

When t = 3, d = 4.9(3)² = 44.1 ✓

When d = 44.1, t = 3 ✓

d(3) = 44.1m, t = 3s for d = 44.1m
Final answer:

An object falls 44.1 meters in 3 seconds. It takes 3 seconds to fall 44.1 meters.

Applied rules:

• Quadratic model: d = 4.9t² describes free fall acceleration

• Solving equations: Substitute known values or solve for unknowns

• Physics context: Model represents gravitational acceleration (g = 9.8 m/s²)

5 Multi-Step Model
Exercise 5
A company's profit is modeled by P(x) = -2x² + 80x - 500, where x is the number of items sold. What is the profit when 20 items are sold? What number of items maximizes profit?
Definition:

Optimization: Finding the maximum or minimum value of a function in a real-world context

P(20)
-2(400) + 80(20) - 500 = 300
Max at vertex
x = -80/(2×-2) = 20
Step 1: Calculate profit when x = 20

Substitute x = 20 into P(x) = -2x² + 80x - 500

P(20) = -2(20)² + 80(20) - 500

P(20) = -2(400) + 1600 - 500 = -800 + 1600 - 500 = 300

Step 2: Find the vertex (maximum point)

For quadratic f(x) = ax² + bx + c, vertex occurs at x = -b/(2a)

Here: a = -2, b = 80, c = -500

x = -80/(2×-2) = -80/(-4) = 20

Step 3: Verify maximum profit

Since a = -2 < 0, the parabola opens downward, so vertex is maximum

Maximum profit occurs when x = 20 items are sold

Step 4: Confirm the maximum profit

P(20) = 300, so maximum profit is $300 when selling 20 items

P(20) = $300, Max at x = 20 items
Final answer:

Profit is $300 when 20 items are sold. The maximum profit of $300 occurs when 20 items are sold.

Applied rules:

• Quadratic optimization: Vertex formula x = -b/(2a) finds extrema

• Direction: Negative coefficient means maximum point

• Business context: Profit model shows relationship between sales and revenue

Mathematical Modeling Analysis Summary
f(x) = ax^2 + bx + c
Quadratic Function
Key definitions:

Mathematical Model: A mathematical representation of a real-world situation using equations, functions, or formulas

Function: A rule that assigns to each input exactly one output

Linear Model: A model of the form f(x) = mx + b, representing constant rate of change

Quadratic Model: A model of the form f(x) = ax² + bx + c, representing parabolic relationships

Exponential Model: A model of the form f(x) = ab^x, representing percent change

Model Validation: The process of checking that a model accurately represents the situation

Domain Restriction: Limiting the input values based on real-world constraints

Optimization: Finding maximum or minimum values in real-world contexts

Complete Mathematical Modeling Process:
  1. Problem analysis: Understand the real-world situation
  2. Variable identification: Choose appropriate input and output variables
  3. Relationship determination: Identify how variables relate to each other
  4. Model selection: Choose appropriate function type (linear, quadratic, exponential)
  5. Parameter determination: Find the constants in the model
  6. Model validation: Check that the model makes sense
  7. Model application: Use the model to solve problems
  8. Result interpretation: Translate mathematical results back to real-world context
Tip 1: Always check if your model makes sense in the real-world context.
Tip 2: Consider domain restrictions based on the problem context.
Tip 3: Use units consistently throughout your model.
Tip 4: Verify your solutions by substituting back into the original problem.
Applications: Used in physics, economics, engineering, biology, and social sciences.
Limitations: Models are simplifications and may not capture all real-world complexities.
Essential Rules:

• Linear model: f(x) = mx + b, constant rate of change

• Quadratic model: f(x) = ax² + bx + c, parabolic behavior

• Exponential model: f(x) = ab^x, percent change

• Vertex formula: x = -b/(2a) for quadratic functions

• Model validation: Check reasonableness and domain restrictions

Questions & Answers

Question: How do I know which type of function to use for modeling a real-world situation?

Answer: Look for key phrases and patterns in the problem:

  • Linear: "constant rate," "per unit," "fixed amount plus variable amount"
  • Quadratic: "area," "square," "increasing/decreasing rate of change"
  • Exponential: "percent increase/decrease," "doubling," "compounding"

Also consider the context:
- Cost models: Often linear (fixed + variable costs)
- Geometric: Often quadratic (area, volume)
- Growth/decay: Often exponential (population, money)

The key is understanding the relationship between the quantities involved.

Question: What if my mathematical model gives an answer that doesn't make sense in the real world?

Answer: This is called model validation, and it's an important part of mathematical modeling:

  • Check your calculations: Make sure there are no arithmetic errors
  • Verify the model: Ensure you chose the right type of function
  • Consider domain restrictions: Some inputs may not make sense
  • Reassess assumptions: Your model may need refinement

For example, if your model predicts negative population, you need to reconsider the domain. If it predicts unrealistic growth, you may need a different model.

Mathematical models are tools that approximate reality, not perfect representations.

Question: How do I find the maximum or minimum of a quadratic function in a real-world context?

Answer: For a quadratic function f(x) = ax² + bx + c, use the vertex formula:

Vertex x-coordinate: x = -b/(2a)

Direction:
- If a > 0: Parabola opens upward (minimum at vertex)
- If a < 0: Parabola opens downward (maximum at vertex)

In real-world contexts:
- Maximum profit, height, area: Use when a < 0
- Minimum cost, distance, time: Use when a > 0

Always interpret the result in the context of the problem.

Question: How do I handle word problems that involve multiple steps or multiple models?

Answer: Break down complex problems systematically:

  1. Read carefully: Identify all the information given
  2. Separate components: Break the problem into smaller parts
  3. Create models: Develop a separate model for each component
  4. Connect models: Use outputs from one model as inputs to another
  5. Solve step-by-step: Work through each component
  6. Combine results: Integrate the partial solutions

For example, if you need to find the total cost of a trip involving different rates for different distances, create separate models for each segment and then add the costs together.

Question: How do I decide what the variables should represent in a real-world problem?

Answer: Follow these guidelines for variable selection:

  • Independent variable: Usually the "input" or "controllable" quantity
  • Dependent variable: Usually the "output" or "result" you want to predict
  • Consider units: Variables should be measurable quantities
  • Think causation: What causes what in the real situation?

For example, in a pricing model, price might be the independent variable (what you set) and demand might be the dependent variable (what you want to predict).

Always clearly define your variables with appropriate units.