Solved Exercises on Modeling Real-World Situations in Grade 7

Master modeling real-world situations: linear functions, proportional relationships, and mathematical modeling through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Distance and Time Model
Exercise 1
A car travels at a constant speed of 60 km/h. Write an equation that models the distance traveled as a function of time. How far will the car travel in 3.5 hours?
Definition:

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

Linear Function: A function with a constant rate of change, represented by y = mx + b.

Constant Speed: When speed remains unchanged, distance is directly proportional to time.

Modeling methodology:
  1. Identify the variables involved
  2. Determine the relationship between variables
  3. Write the mathematical equation
  4. Use the model to make predictions
  5. Verify the solution makes sense
Step 1: Identify variables

Independent variable: Time (t) in hours

Dependent variable: Distance (d) in kilometers

Step 2: Determine the relationship

Distance = Speed × Time

Since speed is constant at 60 km/h, this is a direct proportion

Step 3: Write the equation

d = 60t

This is a linear function where slope = 60 (speed)

Step 4: Calculate distance for 3.5 hours

d = 60 × 3.5 = 210 km

Step 5: Verify the solution

Check: 60 km/h × 3.5 h = 210 km ✓

Final answer:

The equation is d = 60t, and the car will travel 210 km in 3.5 hours.

Applied rules:

• Distance formula: Distance = Rate × Time

• Linear relationship: When rate is constant, distance varies linearly with time

• Direct proportion: As time increases, distance increases at a constant rate

Distance vs Time (60 km/h) Time (hours) Distance (km) 0 1 2 3 4 0 100 200 300 3.5h, 210km
2 Cost Model
Exercise 2
A store sells notebooks for $2.50 each. There is also a fixed shipping fee of $5.00. Write a model for the total cost as a function of the number of notebooks purchased. What is the total cost for 12 notebooks?
Definition:

Linear Model with Intercept: A function of the form y = mx + b, where b is the y-intercept.

Fixed Cost: A cost that remains constant regardless of the quantity.

Variable Cost: A cost that changes with the quantity.

Step 1: Identify variables

Independent variable: Number of notebooks (n)

Dependent variable: Total cost (C) in dollars

Step 2: Identify cost components

Variable cost: $2.50 per notebook

Fixed cost: $5.00 shipping fee

Step 3: Write the equation

Total cost = Variable cost + Fixed cost

C = 2.50n + 5.00

Step 4: Calculate cost for 12 notebooks

C = 2.50(12) + 5.00 = 30.00 + 5.00 = $35.00

Step 5: Verify the solution

Cost of 12 notebooks: 12 × $2.50 = $30.00

Plus shipping: $30.00 + $5.00 = $35.00 ✓

Final answer:

The equation is C = 2.50n + 5.00, and the total cost for 12 notebooks is $35.00.

Applied rules:

• Total cost model: Fixed cost + (Unit price × Quantity)

• Linear function: y = mx + b where m is the rate and b is the intercept

• Break-even analysis: Understanding fixed vs variable costs

Cost vs Number of Notebooks Number of Notebooks Cost ($) 0 5 10 15 20 5 15 25 35 12, $35 $5 intercept
3 Water Tank Model
Exercise 3
A water tank initially contains 200 liters of water. Water flows out at a rate of 15 liters per minute. Write a model for the remaining water as a function of time. How long until the tank is empty?
Definition:

Decay Model: A function that decreases over time at a constant rate.

Initial Value: The starting amount before any change occurs.

Rate of Change: How much the dependent variable changes per unit of the independent variable.

Step 1: Identify variables

Independent variable: Time (t) in minutes

Dependent variable: Water volume (V) in liters

Step 2: Identify initial value and rate

Initial volume: 200 liters

Rate of decrease: 15 liters per minute

Step 3: Write the equation

Volume = Initial volume - (Rate × Time)

V = 200 - 15t

Step 4: Find when tank is empty

Set V = 0 and solve for t

0 = 200 - 15t

15t = 200

t = 200/15 = 13.33 minutes

Step 5: Verify the solution

After 13.33 minutes: V = 200 - 15(13.33) = 200 - 200 = 0 ✓

Final answer:

The equation is V = 200 - 15t, and the tank will be empty after 13.33 minutes.

Applied rules:

• Decay model: y = initial value - (rate × x)

• Linear relationship: Constant rate of change creates a straight line

• Domain restriction: Model is valid only for positive volumes

Water Volume vs Time Time (minutes) Volume (liters) 0 5 10 15 20 0 50 100 150 13.33m, 0L 200L initial
Modeling Concepts, Rules and Methods
y = mx + b
Linear Function Model
Rate of Change
Δy/Δx
Slope of the line
Initial Value
y-intercept
Starting amount
Direct Variation
y = kx
Proportional relationship
Key definitions:

Mathematical Modeling: Creating mathematical representations of real-world situations to understand, predict, or optimize outcomes.

Linear Model: A model that represents a relationship with a constant rate of change, forming a straight line when graphed.

Independent Variable: The input variable that can be controlled or changed (x-axis).

Dependent Variable: The output variable that responds to changes in the independent variable (y-axis).

Modeling methodology:
  1. Problem identification: Understand the real-world situation
  2. Variable selection: Choose appropriate independent and dependent variables
  3. Relationship determination: Identify how variables relate to each other
  4. Equation formulation: Create the mathematical model
  5. Model validation: Check if the model accurately represents the situation
  6. Prediction: Use the model to make forecasts or solve problems
Tip 1: Always identify what the variables represent in the real-world context.
Tip 2: Check if your model makes sense by testing with known values.
Tip 3: Consider the domain - when does your model apply?
Tip 4: Graph your model to visualize the relationship between variables.
Key characteristics: Models should be accurate, predictive, and interpretable in real-world terms.
Common applications: Finance, physics, engineering, economics, and everyday decision making.
Solution: Exercises 4 to 5
4 Population Growth Model
Exercise 4
A small town has a population of 5,000 people. The population grows by 3% each year. Write a model for the population as a function of years from now. What will the population be after 5 years?
Definition:

Exponential Growth Model: A model where the rate of increase is proportional to the current amount.

Growth Factor: The multiplier applied to the current value each period.

Step 1: Identify variables

Independent variable: Time (t) in years

Dependent variable: Population (P)

Step 2: Determine growth parameters

Initial population: 5,000

Growth rate: 3% = 0.03

Growth factor: 1 + 0.03 = 1.03

Step 3: Write the exponential equation

For exponential growth: P = P₀(1 + r)ᵗ

P = 5000(1.03)ᵗ

Step 4: Calculate population after 5 years

P = 5000(1.03)⁵

P = 5000(1.159274) ≈ 5,796 people

Step 5: Verify the calculation

Year 1: 5000 × 1.03 = 5150

Year 2: 5150 × 1.03 = 5304.5

Year 3: 5304.5 × 1.03 = 5463.6

Year 4: 5463.6 × 1.03 = 5627.5

Year 5: 5627.5 × 1.03 = 5796.3 ≈ 5,796 ✓

Final answer:

The equation is P = 5000(1.03)ᵗ, and the population will be approximately 5,796 people after 5 years.

Applied rules:

• Exponential growth: P = P₀(1 + r)ᵗ where r is the growth rate

• Compound growth: Growth builds on previous growth

• Percent conversion: Convert percentage to decimal for calculations

Population Growth (3% annually) Years Population (×1000) 0 2 4 6 8 10 5 6 7 5y, 5.8K
5 Battery Drain Model
Exercise 5
A phone battery starts at 100% charge. It drains at a rate of 8% per hour of usage. Write a model for the battery charge as a function of hours used. How long before the battery reaches 20%?
Definition:

Linear Decay Model: A model that decreases at a constant rate over time.

Percent Decay: A rate of decrease expressed as a percentage per time period.

Step 1: Identify variables

Independent variable: Hours of usage (h)

Dependent variable: Battery charge (B) in percent

Step 2: Identify initial value and decay rate

Initial charge: 100%

Decay rate: 8% per hour

Step 3: Write the linear decay equation

Battery charge = Initial charge - (Decay rate × Time)

B = 100 - 8h

Step 4: Find when battery reaches 20%

Set B = 20 and solve for h

20 = 100 - 8h

8h = 100 - 20

8h = 80

h = 10 hours

Step 5: Verify the solution

After 10 hours: B = 100 - 8(10) = 100 - 80 = 20% ✓

Final answer:

The equation is B = 100 - 8h, and the battery will reach 20% after 10 hours of usage.

Applied rules:

• Linear decay: y = initial value - (rate × x)

• Constant rate: The same amount is lost per unit time

• Domain consideration: Model is valid only while battery charge is positive

Battery Charge vs Usage Time Hours of Usage Charge (%) 0 2 4 6 8 10 0 20 40 60 10h, 20% 100% initial
Modeling Theory: Laws, Methods, Definitions, and Formulas
y = mx + b
Linear Model
Key definitions:

Mathematical Modeling: The process of creating mathematical representations of real-world phenomena to analyze, predict, or optimize outcomes.

Linear Model: A model of the form y = mx + b, where m is the rate of change and b is the initial value.

Exponential Model: A model of the form y = a·bˣ, where b represents the growth or decay factor.

Independent Variable: The input variable (x) that is controlled or manipulated.

Modeling methodology:
  1. Problem understanding: Identify the real-world situation and what needs to be modeled
  2. Variable identification: Determine which quantities change and how they relate
  3. Pattern recognition: Identify the type of relationship (linear, exponential, etc.)
  4. Equation formulation: Create the mathematical model
  5. Model validation: Test the model against known data
  6. Prediction and application: Use the model to make forecasts or solve problems
Tip 1: Always define what each variable represents in the real-world context.
Tip 2: Consider the domain and range of your model - when is it valid?
Tip 3: Graph your model to visualize the relationship and check for reasonableness.
Tip 4: Verify your model with known data points to ensure accuracy.

Key characteristics: Models should be accurate, predictive, interpretable, and applicable to the situation.
Common applications: Economics, physics, biology, business, engineering, and everyday problem solving.
Essential formulas and rules:

• Linear model: y = mx + b (constant rate of change)

• Exponential model: y = a·bˣ (proportional rate of change)

• Direct variation: y = kx (passes through origin)

• Percent change: New value = Old value × (1 ± rate)

• Rate of change: (Change in y) ÷ (Change in x)

Questions & Answers

Question: How do I know if I should use a linear model or an exponential model for a real-world situation?

Answer: The key is to look at the rate of change:

  • Linear model: Use when the rate of change is constant (same amount added/subtracted each time period)
  • Exponential model: Use when the rate of change is proportional to the current amount (same percentage added/subtracted each time period)

Example of linear: Car traveling at constant speed (distance increases by same amount each hour)

Example of exponential: Population growth at fixed percentage rate (population increases by larger amounts over time)

Ask yourself: "Does the quantity change by a fixed amount or by a fixed percentage?"

Question: What's the difference between the slope of a linear model and the rate of change?

Answer: They are actually the same thing in the context of linear models:

  • Slope: The mathematical term for the steepness of a line
  • Rate of change: The real-world interpretation of how quickly the dependent variable changes with respect to the independent variable

In the linear equation y = mx + b, the coefficient m represents both the slope of the line and the rate of change of the situation.

For example, in d = 60t, the slope is 60 and the rate of change is 60 km per hour.

The slope tells us the mathematical relationship, while rate of change tells us what that relationship means in the real world.

Question: How do I know if my mathematical model is good enough for the situation?

Answer: A good model should meet several criteria:

  • Accuracy: It fits known data points reasonably well
  • Plausibility: The relationships make sense in the real world
  • Predictive power: It can make reasonable forecasts
  • Simplicity: It's not unnecessarily complex for the situation

Test your model by checking if it predicts known outcomes correctly.

Consider the context: A simple linear model might be sufficient for short-term predictions, but more complex models might be needed for long-term accuracy.

Always consider the limitations of your model and the range of validity.

Question: What should I do if my model gives unrealistic answers?

Answer: This often indicates a problem with the model's domain or assumptions:

  • Check the domain: Your model may only be valid for certain input values
  • Re-examine assumptions: The real-world situation might have changed
  • Consider limitations: All models are simplifications of reality

Example: A linear model predicting population growth might give negative values for early years, which is unrealistic.

In such cases, restrict the domain of your model to realistic values or consider a different model type.

Always think about whether your answers make sense in the real-world context.

Question: How can I use mathematical models to make decisions in real life?

Answer: Mathematical models are incredibly useful for real-life decision making:

  • Budgeting: Model expenses vs income to plan spending
  • Travel planning: Model time vs distance to optimize routes
  • Investment decisions: Model growth of savings over time
  • Resource management: Model consumption vs availability

Start by identifying the key variables in your decision, then create a simple model to explore different scenarios.

Example: If you're saving money, create a model of your savings over time considering your income and expenses.

Models help you understand the consequences of different choices before making them.