Growth and decay models
Objective. Build, solve and evaluate continuous and discrete models.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Growth and decay models use exponential functions to describe quantities that change by a constant percentage over equal time intervals, such as P = P0×bᵗ where P0 is the initial value and b is the growth or decay factor per unit time. If a value grows by 8% each year, b = 1.08, and if it decays by 8%, b = 0.92. For example a £2000 investment growing at 5% yearly gives P = 2000×1.05ᵗ, so after 3 years P = 2000×1.05³ ≈ £2315.25. Logarithms are used to find the time needed to reach a target value by solving the resulting exponential equation for t.
Worked example
A radioactive sample decays according to the growth and decay model N = 800×0.9ᵗ, where t is in days. Find how many days it takes for the sample to fall to 400.
- 1Set N = 400: 400 = 800×0.9ᵗ.
- 2Divide both sides by 800: 0.5 = 0.9ᵗ.
- 3Take logs: t = ln(0.5)/ln(0.9).
- 4Compute: t = (−0.6931)/(−0.1054) ≈ 6.58 days.
Answer: t ≈ 6.58 days
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students often use b = 1.8 instead of 1.08 for an 8% increase, forgetting that the growth factor is 1 plus the decimal rate, not the decimal rate itself expressed as a whole number.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.