Exponential growth and decay
Exponential growth and decay (Extended)
- When a quantity changes by the same percentage each period, it grows or decays exponentially.
- Use the compound formula: value $= P\left(1 + \dfrac{r}{100}\right)^t$ (a negative rate for decay).
- Depreciation (e.g. a car losing value) is decay.
Practice
Depreciation of a car's value each year is an example of:
Losing a fixed percentage each year is exponential decay.
Example
- A car worth 20 000 loses 15% each year (multiplier 0.85):
- after 4 years: $20\,000 \times 0.85^4 \approx 10\,440$.
Practice
A car loses 15% of its value each year. What yearly multiplier do you use?
Losing 15% leaves 85%, so the multiplier is 0.85.
Practice
A car worth 20000 depreciates 15% per year. Its value after 4 years (20000 × 0.85^4) is about (nearest 10)?
20000 × 0.85^4 = 20000 × 0.5220 ≈ 10440.
You've got it
Key idea
- exponential change uses the compound formula $P(1 + r/100)^t$
- growth = positive rate; decay/depreciation = a multiplier below 1
- a 15% yearly loss uses the multiplier 0.85