Grade: Grade 12 Subject: Mathematics Unit: Precalculus Completion Lesson: 4 of 6 SAT: AdvancedMath ACT: Math

Word Problems

Apply precalculus concepts to solve real-world problems involving growth, decay, finance, and physical phenomena.

Learn

Word problems connect abstract mathematical concepts to practical situations. This lesson focuses on translating verbal descriptions into mathematical models using limits, sequences, and series.

Types of Real-World Applications

  • Population Growth: Modeled using geometric sequences and exponential limits
  • Financial Applications: Compound interest, annuities, and loan payments use geometric series
  • Physics Applications: Velocity, acceleration, and distance involve limits and sequences
  • Decay Problems: Radioactive decay, depreciation, and drug concentration use exponential models

Problem-Solving Framework

  1. Read Carefully: Identify what is given and what is being asked
  2. Define Variables: Assign variables to unknown quantities
  3. Identify the Pattern: Determine if the situation is arithmetic, geometric, or involves limits
  4. Set Up the Equation: Translate the verbal description into mathematical notation
  5. Solve and Verify: Compute the answer and check it makes sense in context

Key Formulas for Applications

  • Compound Interest: A = P(1 + r/n)^(nt)
  • Continuous Compounding: A = Pe^(rt)
  • Annuity Future Value: FV = PMT * [(1 + r)^n - 1]/r
  • Annuity Present Value: PV = PMT * [1 - (1 + r)^(-n)]/r
  • Exponential Growth/Decay: A(t) = A_0 * e^(kt)

Examples

Study these worked examples to understand how to approach word problems.

Example 1: Population Growth

Problem: A bacteria colony starts with 500 bacteria and doubles every 4 hours. How many bacteria will there be after 24 hours?

Solution:

Step 1: This is geometric growth with a_1 = 500 and r = 2

Step 2: Number of doubling periods: 24/4 = 6

Step 3: After n doublings: a_n = 500 * 2^6 = 500 * 64 = 32,000

Answer: There will be 32,000 bacteria after 24 hours.

Example 2: Compound Interest

Problem: You deposit $5,000 in an account that earns 6% interest compounded monthly. What will the account balance be after 10 years?

Solution:

Step 1: P = 5000, r = 0.06, n = 12, t = 10

Step 2: A = 5000(1 + 0.06/12)^(12*10)

Step 3: A = 5000(1.005)^120

Step 4: A = 5000(1.8194) = $9,097.01

Answer: The account balance will be approximately $9,097.01.

Example 3: Bouncing Ball (Infinite Series)

Problem: A ball is dropped from 16 feet and bounces to 75% of its previous height each time. Find the total vertical distance traveled by the ball.

Solution:

Step 1: First drop = 16 ft (down only)

Step 2: Each bounce: up and down, so multiply by 2

Step 3: Bounces form geometric series: 2(12) + 2(9) + 2(6.75) + ... = 2 * [12 + 9 + 6.75 + ...]

Step 4: Sum of bounce heights: 12/(1 - 0.75) = 12/0.25 = 48

Step 5: Total = 16 + 2(48) = 16 + 96 = 112 feet

Answer: The ball travels a total of 112 feet.

Practice

Solve these real-world problems. Show your work and include units in your answers.

Problem 1: A car depreciates in value by 15% each year. If the car is worth $28,000 new, what will it be worth after 5 years?

Problem 2: A salary starts at $45,000 and increases by $2,500 each year. What is the total amount earned over 8 years?

Problem 3: You invest $200 per month in a retirement account that earns 7% annual interest compounded monthly. How much will you have after 30 years?

Problem 4: A radioactive substance has a half-life of 5 days. If you start with 80 grams, how much remains after 20 days?

Problem 5: The population of a city is 50,000 and grows at 3% per year. Use the limit concept to find the population after many years if growth becomes continuous at this rate. (Hint: Use continuous growth formula)

Problem 6: A theater has 20 seats in the first row, 24 seats in the second row, and 28 seats in the third row, continuing the pattern for 15 rows total. How many total seats are in the theater?

Problem 7: A drug has a half-life of 6 hours in the body. If a patient takes a 400mg dose, what amount remains after 24 hours?

Problem 8: You take out a $200,000 mortgage at 4.5% annual interest compounded monthly for 30 years. What is your monthly payment? (Use the present value annuity formula solved for PMT)

Problem 9: A pendulum swings 100 cm on its first swing and 85% of the previous distance on each subsequent swing. What is the total distance the pendulum travels before coming to rest?

Problem 10: An athlete runs 10 km on the first day of training and increases distance by 0.5 km each day. On which day will they first run at least 20 km?

Problem 11: A company's profit was $2 million in year 1 and grows by 8% each year. What is the total profit over the first 10 years?

Problem 12: Water is draining from a tank. Each minute, 20% of the remaining water drains. If the tank starts with 500 gallons, what is the limit of water remaining as time approaches infinity?

Check Your Understanding

Compare your answers and methods with the solutions below.

Answer Key

  1. V = 28000(0.85)^5 = $12,435.65
  2. a_1 = 45000, d = 2500, n = 8; S_8 = 8(45000 + 62500)/2 = $430,000
  3. FV = 200 * [(1 + 0.07/12)^360 - 1]/(0.07/12) = $243,994.47
  4. 20 days = 4 half-lives; 80 * (1/2)^4 = 5 grams
  5. Using A = Pe^(rt), continuous growth approaches infinity; however, for any finite time t: A(t) = 50000e^(0.03t)
  6. a_1 = 20, d = 4, n = 15; S_15 = 15(20 + 76)/2 = 720 seats
  7. 24 hours = 4 half-lives; 400 * (1/2)^4 = 25 mg
  8. PMT = 200000 * (0.045/12)/[1 - (1 + 0.045/12)^(-360)] = $1,013.37/month
  9. Total distance = 100 + 2 * [85/(1-0.85)] = 100 + 2(566.67) = 1,233.33 cm
  10. 10 + (n-1)(0.5) >= 20; n >= 21; Day 21
  11. S = 2,000,000 * [(1.08)^10 - 1]/0.08 = $28,973,123.52
  12. As each minute 80% remains, lim = 500 * 0 = 0 gallons

Next Steps

  • Review the problem types you found most challenging
  • Practice setting up equations before solving
  • Proceed to Common Mistakes to learn how to avoid typical errors
  • Create your own word problems based on real situations you encounter