Growth and decay · Grade 12
In Grade 12 you get fluent with: exponential models.
When something grows or shrinks by the same percentage in each time period, it follows an exponential model N = a × bᵗ. Populations, bacteria, radioactive material, medicine in the blood and the value of a car all behave this way.
Key facts
- N = a × bᵗ: a is the starting value and b is the multiplier for each time period.
- b > 1 means growth: b = 1.04 is +4% each period. 0 < b < 1 means decay: b = 0.9 is −10% each period.
- Half-life is the time for an amount to halve. After n half-lives, amount = start × (1/2)ⁿ.
- To find a time, use logarithms: bᵗ = k gives t = log k ÷ log b.
- Continuous growth: N = N₀e^(kt). Find k with natural logs (ln).
- Multiplier from data: b = (later ÷ earlier)^(1/n), where n is the number of periods.
Worked examples
Example 1
A town of 12,000 people grows by 3% a year. Find the population after 5 years.
Answer: 13,911
Example 2
A population grows from 5,000 to 6,500 in 6 years. Find the yearly growth rate, to 1 decimal place.
Answer: 4.5%