Grade: Grade 12 Subject: Mathematics Unit: Real-World Modeling Lesson: 3 of 6 SAT: ProblemSolving+DataAnalysis ACT: Math

Guided Practice

Learn

In this guided practice session, you will work through structured problems that apply the mathematical modeling concepts from the previous lessons. Each problem includes step-by-step guidance to help you develop confidence before tackling problems independently.

The Modeling Process Review

Before diving into practice, let's review the key steps in mathematical modeling:

  1. Identify the problem - What question are we trying to answer?
  2. Make assumptions - What simplifications are reasonable?
  3. Define variables - What quantities are we measuring?
  4. Build the model - What mathematical relationships exist?
  5. Solve and interpret - What does the solution tell us?
  6. Validate - Does our answer make sense in context?

Types of Models You Will Practice

  • Linear models - Constant rate of change situations
  • Exponential models - Growth and decay scenarios
  • Quadratic models - Projectile motion, optimization
  • Piecewise models - Different rules for different conditions

Examples

Example 1: Population Growth Model

Problem: A city's population was 50,000 in 2020 and grows at 2.5% per year. Model the population and predict when it will reach 75,000.

Step 1: Identify the model type. This is exponential growth with rate r = 0.025.

Step 2: Write the model: P(t) = 50,000(1.025)^t, where t = years since 2020.

Step 3: Solve for t when P(t) = 75,000:

75,000 = 50,000(1.025)^t

1.5 = (1.025)^t

ln(1.5) = t * ln(1.025)

t = ln(1.5)/ln(1.025) = 16.4 years

Answer: The population will reach 75,000 around mid-2036.

Example 2: Cost Optimization

Problem: A company's profit P (in thousands) is modeled by P(x) = -2x^2 + 120x - 1000, where x is units sold (in hundreds). Find the number of units that maximizes profit.

Step 1: Recognize this as a downward parabola (a = -2 < 0).

Step 2: Find the vertex using x = -b/(2a):

x = -120/(2 * -2) = 30

Step 3: Convert units: 30 hundreds = 3,000 units

Step 4: Find maximum profit: P(30) = -2(900) + 3600 - 1000 = $800,000

Practice Problems

Work through each problem using the modeling process. Solutions are provided at the end.

Problem 1: Depreciation Model

A car purchased for $35,000 depreciates at 15% per year. Write a model for the car's value V after t years. What will the car be worth after 5 years?

Problem 2: Temperature Conversion

The relationship between Celsius (C) and Fahrenheit (F) is linear. Water freezes at 0 C (32 F) and boils at 100 C (212 F). Write a model to convert C to F, and find what temperature is the same in both scales.

Problem 3: Projectile Height

A ball is thrown upward with an initial velocity of 64 ft/s from a height of 48 ft. The height h(t) = -16t^2 + 64t + 48. When does the ball reach its maximum height, and what is that height?

Problem 4: Investment Comparison

Investment A offers 5% simple interest. Investment B offers 4.5% compounded annually. Write models for both and determine which is better for a 10-year investment of $10,000.

Problem 5: Break-Even Analysis

A company has fixed costs of $50,000 and variable costs of $12 per unit. They sell each unit for $20. Write a profit model and find the break-even point.

Problem 6: Mixture Problem

How many liters of a 20% acid solution must be mixed with 30 liters of a 60% acid solution to create a 35% acid solution? Write an equation and solve.

Problem 7: Cooling Model

Coffee at 180 F is left in a room at 70 F. Using Newton's Law of Cooling, T(t) = 70 + 110e^(-0.05t), find when the coffee reaches 100 F.

Problem 8: Distance-Rate-Time

Two cars leave the same point at the same time, one traveling north at 55 mph and one traveling east at 40 mph. Write a model for the distance between them as a function of time, and find the distance after 2 hours.

Problem 9: Revenue Maximization

A theater finds that at a price of $12, they sell 400 tickets. For each $1 increase, they sell 25 fewer tickets. Write a revenue model and find the price that maximizes revenue.

Problem 10: Compound Interest

How long will it take for an investment to triple if it earns 6% interest compounded continuously? Use the formula A = Pe^(rt).

Check Your Understanding

Review these key concepts before moving on:

  1. Can you identify whether a scenario calls for a linear, exponential, or quadratic model?
  2. Are you comfortable using logarithms to solve exponential equations?
  3. Can you find the vertex of a parabola to solve optimization problems?
  4. Do you validate your answers by checking if they make sense in context?

Answer Key

  1. V(t) = 35,000(0.85)^t; After 5 years: approximately $15,537
  2. F = (9/5)C + 32; Same temperature: -40
  3. Maximum at t = 2 seconds; Height = 112 feet
  4. A: $15,000; B: $15,530; Investment B is better
  5. P(x) = 8x - 50,000; Break-even at 6,250 units
  6. 50 liters of 20% solution
  7. Approximately 19.8 minutes
  8. d(t) = sqrt((55t)^2 + (40t)^2) = t*sqrt(4625); After 2 hours: 136 miles
  9. R(x) = (12 + x)(400 - 25x); Maximum at $14 per ticket
  10. Approximately 18.3 years

Next Steps

  • Practice identifying model types from word problem descriptions
  • Work on translating verbal descriptions into mathematical equations
  • Review logarithm properties if exponential equations were challenging
  • Move on to Word Problems for more independent practice