6.8 Exponential Growth and Decay

Exponential growth and decay models are powerful tools for understanding real-world phenomena. They describe how quantities change over time, whether it's population growth, compound interest, or radioactive decay.

These models use simple equations to predict complex behaviors. By understanding doubling time and half-life, we can make accurate predictions about future quantities in various fields, from finance to physics.

Exponential Growth Models

Applications of exponential growth

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• Exponential growth involves quantities increasing by constant percentage over equal time intervals
  • General exponential growth function: $y = a(1 + r)^t$
    • $a$: initial amount or population (base)
    • $r$: growth rate per time interval (decimal)
    • $t$: number of time intervals
• Population dynamics often modeled with exponential growth
  • Unconstrained populations grow exponentially
  • Bacteria starting at 100 cells, doubling hourly: $P(t) = 100 \cdot 2^t$ after $t$ hours
• Compound interest calculations use exponential growth
  • Principal grows exponentially when interest compounded
  • Compound interest formula: $A = P(1 + \frac{r}{n})^{nt}$
    • $A$: final amount
    • $P$: initial principal
    • $r$: annual interest rate (decimal)
    • $n$: compounding periods per year
    • $t$: number of years

Doubling time in growth models

• Doubling time: time for quantity to double in size
• Calculate doubling time: $t_d = \frac{\ln 2}{k}$
  • $t_d$: doubling time
  • $k$: continuous growth rate (rate constant)
• In exponential growth, doubling time stays constant
  • Time to double remains same regardless of current quantity size

Exponential Decay Models

Uses of exponential decay

• Exponential decay involves quantities decreasing by constant percentage over equal time intervals
  • General exponential decay function: $y = a(1 - r)^t$
    • $a$: initial amount
    • $r$: decay rate per time interval (decimal)
    • $t$: number of time intervals
• Radioactive decay follows exponential decay model
  • Amount of radioactive substance decreases exponentially over time
  • Decay formula: $A(t) = A_0 e^{-\lambda t}$
    • $A(t)$: amount of substance at time $t$
    • $A_0$: initial amount of substance
    • $\lambda$: decay constant
• Temperature change modeled by Newton's Law of Cooling (exponential decay)
  • Object's temperature changes exponentially, approaching ambient temperature
  • Newton's Law of Cooling formula: $T(t) = T_a + (T_0 - T_a)e^{-kt}$
    • $T(t)$: temperature at time $t$
    • $T_a$: ambient temperature (asymptote)
    • $T_0$: initial object temperature
    • $k$: cooling constant

Half-life in decay processes

• Half-life: time for quantity to reduce to half its initial value
• Calculate half-life: $t_{1/2} = \frac{\ln 2}{\lambda}$
  • $t_{1/2}$: half-life
  • $\lambda$: decay constant
• In exponential decay, half-life remains constant
  • Time to halve stays same regardless of current quantity size
• Half-life especially important in radioactive decay
  • Determines age of radioactive materials
  • Predicts remaining amount of radioactive substance after certain time

Mathematical Foundations

• Exponential functions are solutions to certain differential equations
• Logarithmic scale is often used to visualize exponential growth or decay
• Exponential models can be linearized using logarithms for easier analysis