Sample Homework Assignment

Exponential Growth and Decay

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Material Overview

Modeling population growth, compound interest, and half-life

Subject Mathematics
Subject Area Algebra II

Grade/Level 10th Grade

Type Homework Assignment

Difficulty Challenging

Learning Setting Homeschool

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Teacher Context

Teacher Lesson Scope

What You Should Know

Exponential functions model processes where a quantity grows or decays at a rate proportional to its current value. Unlike linear functions, which change by a constant additive amount per unit of time, exponential functions change by a constant multiplicative factor. These models are essential for understanding real-world systems like population growth, compound interest in finance, and the decay of radioactive substances.

Key Skills

  • Distinguish between linear and exponential relationships based on descriptions or data.
  • Write and evaluate exponential growth and decay models, including doubling time and half-life.
  • Calculate compound interest using various compounding frequencies.
  • Interpret the meaning of parameters (initial value, base, and rate) within exponential equations.

Important Vocabulary

  • Exponential Growth: A process where a quantity increases by a constant percentage rate over equal time intervals.
  • Exponential Decay: A process where a quantity decreases by a constant percentage rate over equal time intervals.
  • Growth/Decay Factor: The base of the exponential function, represented as (1 + r) for growth or (1 - r) for decay, where r is the percent rate of change as a decimal.
  • Compound Interest: Interest calculated on both the initial principal and the accumulated interest from previous periods.
  • Half-Life: The amount of time required for a quantity to decrease to half of its initial value.

Assessment Boundaries

This assignment covers writing, evaluating, and interpreting exponential models of the form y = a × bˣ. Financial problems will use compounding periods (such as annual, semi-annual, quarterly, and monthly) using U.S. dollars. Scientific problems will focus on half-life decay, and population models will cover growth, decay, and doubling. You will be expected to perform calculations with precision and round your answers to the specified decimal places.

Printable Student Copy

Student Version

Name:
Date:

Welcome to Modeling Population Growth, Compound Interest, and Half-Life! This homework assignment is designed to challenge your understanding of exponential growth and decay models, compound interest, and half-life applications. Please read each question carefully, show your work for the short-answer questions, and pay close attention to rounding instructions.

  1. A city's population in 2020 was 45,000. It is projected to grow exponentially at a rate of 3.2% per year. Which equation models the population, P(t), t years after 2020, and what is the projected population in 2035 (rounded to the nearest whole number)?
    A) P(t) = 45,000 × (1.032)ᵗ; 72,179
    B) P(t) = 45,000 × (1.32)ᵗ; 4,374,228
    C) P(t) = 45,000 + 1,440t; 66,600
    D) P(t) = 45,000 × (0.968)ᵗ; 27,564

  2. You deposit $2,500 into an account that earns an annual interest rate of 4.5% compounded monthly. How much money will be in the account after 8 years? (Round your final answer to the nearest cent.)
    A) $3,400.00
    B) $3,580.91
    C) $3,576.25
    D) $3,555.25

  3. An isotope of a radioactive element has a half-life of 24 hours. If a scientist starts with a 120-gram sample, which expression represents the amount of the isotope remaining after d days, and how many grams remain after 5 days?
    A) A(d) = 120 × (0.5)^(d/24); 103.92 grams
    B) A(d) = 120 × (0.5)ᵈ; 3.75 grams
    C) A(d) = 120 × (0.5)^(24d); 0.00 grams
    D) A(d) = 120 - 24d; 0 grams

  4. Which of the following scenarios describes a quantity that changes exponentially?
    A) A rental car company charges a flat fee of $45 plus $0.15 for every mile driven.
    B) A local charity receives a donation of $1,500 every month from a regular donor.
    C) The value of a certain model of smartphone depreciates by 18% of its current value each year.
    D) A training athlete increases their daily running distance by exactly 0.5 miles each week.

  5. The population of a rare species of bird on an isolated island is modeled by the function P(t) = 850 × (0.88)ᵗ, where t represents the number of years since a conservation study began. Which of the following is the correct interpretation of the parameters in this model?
    A) The initial population was 850 birds, and the population is increasing by 12% each year.
    B) The initial population was 850 birds, and the population is decreasing by 12% each year.
    C) The population decreases by 88 birds each year, starting from an initial population of 850.
    D) The initial population was 748 birds, and the population is decreasing by 88% each year.

  6. A certain culture of bacteria doubles in size every 4 hours. If the culture starts with 500 bacteria, write an exponential model, N(t), representing the number of bacteria after t hours. Use your model to determine how many bacteria will be present after 18 hours. Round your final answer to the nearest whole number.

  1. You invest $5,000 in a savings account with an annual interest rate of 6.2% compounded quarterly.
    Part A: Write a function, A(t), that models the value of the investment after t years.
    Part B: Determine the account balance after 12 years. Round your final answer to the nearest cent.
  1. The radioactive isotope Iodine-131 has a half-life of approximately 8 days. A medical facility stores a 64-milligram sample.
    Part A: Write an exponential decay function, M(t), that models the remaining mass of Iodine-131 in milligrams after t days.
    Part B: How much of the sample will remain after 20 days? Round your final answer to the nearest hundredth of a milligram.
  1. Suppose you are offered two different incentive plans at a new job. Plan A offers a starting salary of $40,000 with a guaranteed raise of $2,500 each year. Plan B offers a starting salary of $40,000 with a guaranteed 5% salary increase each year.
    Part A: Write salary models, S_A(t) and S_B(t), for both plans after t years.
    Part B: Compare the salaries under both plans at t = 10 years. Show your calculations and state which plan yields a higher salary and by how much (rounded to the nearest dollar).
  1. During your self-study, you have completed various readings, notes, prior practice exercises, or personal observations related to exponential functions. Select one specific example or observation from your self-study (such as a practice problem you solved, a real-world article you read, or notes you took on exponential trends) and briefly explain how it connects to the population growth, compound interest, or half-life concepts practiced in this homework assignment.

Teacher Copy

Answer Key

  1. A city's population in 2020 was 45,000. It is projected to grow exponentially at a rate of 3.2% per year. Which equation models the population, P(t), t years after 2020, and what is the projected population in 2035 (rounded to the nearest whole number)?
    A) P(t) = 45,000 × (1.032)ᵗ; 72,179
    B) P(t) = 45,000 × (1.32)ᵗ; 4,374,228
    C) P(t) = 45,000 + 1,440t; 66,600
    D) P(t) = 45,000 × (0.968)ᵗ; 27,564

  2. You deposit $2,500 into an account that earns an annual interest rate of 4.5% compounded monthly. How much money will be in the account after 8 years? (Round your final answer to the nearest cent.)
    A) $3,400.00
    B) $3,580.91
    C) $3,576.25
    D) $3,555.25

  3. An isotope of a radioactive element has a half-life of 24 hours. If a scientist starts with a 120-gram sample, which expression represents the amount of the isotope remaining after d days, and how many grams remain after 5 days?
    A) A(d) = 120 × (0.5)^(d/24); 103.92 grams
    B) A(d) = 120 × (0.5)ᵈ; 3.75 grams
    C) A(d) = 120 × (0.5)^(24d); 0.00 grams
    D) A(d) = 120 - 24d; 0 grams

  4. Which of the following scenarios describes a quantity that changes exponentially?
    A) A rental car company charges a flat fee of $45 plus $0.15 for every mile driven.
    B) A local charity receives a donation of $1,500 every month from a regular donor.
    C) The value of a certain model of smartphone depreciates by 18% of its current value each year.
    D) A training athlete increases their daily running distance by exactly 0.5 miles each week.

  5. The population of a rare species of bird on an isolated island is modeled by the function P(t) = 850 × (0.88)ᵗ, where t represents the number of years since a conservation study began. Which of the following is the correct interpretation of the parameters in this model?
    A) The initial population was 850 birds, and the population is increasing by 12% each year.
    B) The initial population was 850 birds, and the population is decreasing by 12% each year.
    C) The population decreases by 88 birds each year, starting from an initial population of 850.
    D) The initial population was 748 birds, and the population is decreasing by 88% each year.

  6. A certain culture of bacteria doubles in size every 4 hours. If the culture starts with 500 bacteria, write an exponential model, N(t), representing the number of bacteria after t hours. Use your model to determine how many bacteria will be present after 18 hours. Round your final answer to the nearest whole number.

  7. You invest $5,000 in a savings account with an annual interest rate of 6.2% compounded quarterly.
    Part A: Write a function, A(t), that models the value of the investment after t years.
    Part B: Determine the account balance after 12 years. Round your final answer to the nearest cent.

  8. The radioactive isotope Iodine-131 has a half-life of approximately 8 days. A medical facility stores a 64-milligram sample.
    Part A: Write an exponential decay function, M(t), that models the remaining mass of Iodine-131 in milligrams after t days.
    Part B: How much of the sample will remain after 20 days? Round your final answer to the nearest hundredth of a milligram.

  9. Suppose you are offered two different incentive plans at a new job. Plan A offers a starting salary of $40,000 with a guaranteed raise of $2,500 each year. Plan B offers a starting salary of $40,000 with a guaranteed 5% salary increase each year.
    Part A: Write salary models, S_A(t) and S_B(t), for both plans after t years.
    Part B: Compare the salaries under both plans at t = 10 years. Show your calculations and state which plan yields a higher salary and by how much (rounded to the nearest dollar).

  10. During your self-study, you have completed various readings, notes, prior practice exercises, or personal observations related to exponential functions. Select one specific example or observation from your self-study (such as a practice problem you solved, a real-world article you read, or notes you took on exponential trends) and briefly explain how it connects to the population growth, compound interest, or half-life concepts practiced in this homework assignment.

Answers and Explanations

  1. A) P(t) = 45,000 × (1.032)ᵗ; 72179
    Explanation: An exponential growth model has the form P(t) = P_0 × (1 + r)ᵗ. Here, P_0 = 45,000 and r = 0.032, so the model is P(t) = 45,000 × (1.032)ᵗ. The year 2035 is t = 15 years after 2020. Substituting t = 15 yields a projected population of 72179.

  2. B) $3580.91
    Explanation: Using the compound interest formula A = P × (1 + r/n)^(n × t) with P = 2500, r = 0.045, n = 12, and t = 8, we find the account balance after 8 years is $3580.91.

  3. B) A(d) = 120 × (0.5)ᵈ; 3.75 grams
    Explanation: Since the half-life is 24 hours, which is exactly 1 day, the remaining amount after d days is modeled by A(d) = 120 × (0.5)^d. After 5 days, A(5) = 120 × (0.5)⁵ = 3.75 grams.

  4. C) The value of a certain model of smartphone depreciates by 18% of its current value each year.
    Explanation: A constant percentage decrease represents exponential decay, whereas constant absolute changes represent linear growth or decay.

  5. B) The initial population was 850 birds, and the population is decreasing by 12% each year.
    Explanation: The value 850 is the initial population. The growth factor is 0.88, which represents a decrease of 1 - 0.88 = 0.12, or 12% per year.

  6. N(t) = 500 × (2)^(t/4); 11,314 bacteria
    Explanation: The exponential growth model is N(t) = 500 × 2^(t/4). For t = 18 hours, N(18) = 500 × 2^(184) = 500 × 2⁴.5 ≈ 11314 bacteria.

  7. Part A: A(t) = 5,000 × (1.0155)^(4t); Part B: $10461.80
    Explanation: Part A: The quarterly compounding formula is A(t) = P × (1 + r/4)^(4t). With P = 5,000 and r = 0.062, the rate per quarter is 0.062/4 = 0.0155, leading to the model A(t) = 5,000 × (1.0155)^(4t). Part B: Substituting t = 12 years into the model yields a balance of $10461.80.

  8. Part A: M(t) = 64 × (0.5)^(t/8); Part B: 11.31 milligrams
    Explanation: Part A: The half-life is 8 days, so M(t) = 64 × (0.5)^(t/8). Part B: M(20) = 64 × (0.5)^(208) = 64 × (0.5)².5 ≈ 11.31 milligrams.

  9. Part A: S_A(t) = 40,000 + 2,500t, S_B(t) = 40,000 × (1.05)ᵗ; Part B: Plan A yields $65,000, Plan B yields $65,156. Plan B is higher by $156.
    Explanation: At t = 10, Plan A: S_A(10) = 40000 + 2500 × 10 = 65000. Plan B: S_B(10) = 40000 × (1.05)¹⁰ ≈ 65156. The difference is 65156 - 65000 = 156.

  10. Answers will vary. A complete response must select one specific self-study example and connect it clearly to exponential growth, decay, or compound interest.
    Explanation: This is an open-ended reflection question requiring students to link personal self-study materials to course topics.