M5.2.6 For the adult
Growth and decay models
A herd of goats grows by 8% every year. How do we write a rule for its size, and when does such a rule stop being true?
What this chapter teaches
A quantity that grows or decays by a fixed percentage each step follows y = A(1 + r) to the power t, or equally y = A exp(kt) with k = ln (1 + r). The doubling time is ln 2 ÷ k, and the half-life of a decay is ln 2 ÷ |k|. A model is trusted only near the data it was built from.
Three questions to ask
There is no one right wording. Listen for the idea.
- Why is growth of 8% a year a factor of 1.08, and not 8 or 0.08?
- If a sample loses half its mass in 5 days, how much is left after 10 days, and after 15 days?
- Why can the herd model not be right after 100 years, even if it fits the first 5 years well?
After the chapter, the learner can
- Model a goat herd growing by 8% a year with an exponential
- Find a half life from a decay model
- Say when an exponential model stops making sense
What they copy into their notebook
Growth by r: y = A(1 + r) to the power t. Also y = A exp(kt), with k = ln (1 + r). Doubling time ln 2 ÷ k. Half-life ln 2 ÷ |k|. Trust a model near its data only.