and decay models are powerful tools for understanding real-world phenomena. They describe how quantities change over time, whether it's , , or radioactive decay.
These models use simple equations to predict complex behaviors. By understanding and , 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⋅2t after t hours
Compound interest calculations use exponential growth
Principal grows exponentially when interest compounded
Compound interest formula: A=P(1+nr)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: td=kln2
td: 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
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)=A0e−λt
A(t): amount of substance at time t
A0: initial amount of substance
λ: 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)=Ta+(T0−Ta)e−kt
T(t): temperature at time t
Ta: ambient temperature (asymptote)
T0: 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: t1/2=λln2
t1/2: half-life
λ: 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