Exponential growth and decay describe situations where a quantity changes by a constant percent over equal time intervals. This idea appears in compound interest, bacteria populations, radioactive materials, cooling objects, and medicine levels in the body. A strong school project can use real or simulated data to show how a simple equation becomes a graph and a prediction tool.
The main goal is to connect the pattern in a table, the shape of a curve, and the meaning of the parameters.
Key Facts
- General exponential model: y = a e^(kt), where a is the initial amount, k is the rate constant, and t is time.
- Growth has k > 0, so the graph rises faster over time.
- Decay has k < 0, so the graph decreases toward zero but does not usually reach zero.
- Percent change model: y = a(1 + r)^t for growth and y = a(1 - r)^t for decay, when r is the rate per time step.
- Half-life formula: t1/2 = ln(2)/|k| for the continuous decay model y = a e^(kt).
- Doubling time formula for growth: td = ln(2)/k when k > 0.
Vocabulary
- Exponential growth
- A pattern where a quantity increases by the same percent during each equal time interval.
- Exponential decay
- A pattern where a quantity decreases by the same percent during each equal time interval.
- Rate constant
- The value k in y = a e^(kt) that controls how quickly the exponential model grows or decays.
- Half-life
- The time required for a decaying quantity to fall to half of its current amount.
- Initial amount
- The starting value a of the quantity when time t = 0.
Common Mistakes to Avoid
- Treating exponential change like linear change is wrong because exponential models change by a constant percent, not a constant amount.
- Using a positive k for decay is wrong because k must be negative in y = a e^(kt) when the quantity decreases over time.
- Confusing half-life with total lifetime is wrong because half-life tells how long it takes to lose half of the current amount, not when the amount becomes zero.
- Forgetting units on time and rate is wrong because k must match the time unit used in the data, such as per hour or per day.
Practice Questions
- 1 A bacteria culture starts with 500 cells and grows according to y = 500e^(0.35t), where t is in hours. Find the population after 6 hours, rounded to the nearest whole cell.
- 2 A radioactive sample starts with 80 grams and follows y = 80e^(-0.12t), where t is in years. Find the half-life and the amount remaining after 10 years.
- 3 A student collects data for a cooling cup of water and notices the temperature difference from room temperature is multiplied by about 0.75 every 5 minutes. Explain why an exponential decay model is more appropriate than a linear model.