Exponential Decay Applications: From Radioactive Decay to Depreciation

Understanding Exponential Decay 📉

Exponential decay describes the decrease in a quantity over time. It's characterized by a rate of decrease proportional to the current amount. The general formula for exponential decay is:

N(t) = N_0 * e^{-λt}

Where:

• $N(t)$ is the quantity at time $t$,
• $N_0$ is the initial quantity,
• $λ$ (lambda) is the decay constant,
• $e$ is the base of the natural logarithm (approximately 2.71828).

Radioactive Decay ☢️

Radioactive decay is a classic example. Unstable atomic nuclei lose energy by emitting radiation. The half-life ($t_{1/2}$) is the time it takes for half of the substance to decay.

The relationship between the decay constant ($λ$) and half-life is:

t_{1/2} = ln(2) / λ

Example: Carbon-14 dating.

import math

N0 = 100  # Initial amount of Carbon-14
t = 5730  # Half-life of Carbon-14 in years
lambda_val = math.log(2) / t  # Decay constant
time = 10000 # Time elapsed

Nt = N0 * math.exp(-lambda_val * time)
print(f"Amount of Carbon-14 remaining after {time} years: {Nt:.2f}")

Depreciation of Assets 💰

Depreciation refers to the reduction in the value of an asset over time. While several depreciation methods exist, exponential decay can model certain types of depreciation.

Let's consider an asset losing value exponentially:

V(t) = V_0 * e^{-kt}

• $V(t)$ is the value of the asset at time $t$,
• $V_0$ is the initial value,
• $k$ is the depreciation rate.

Example: A car's value depreciating over time.

import math

V0 = 25000  # Initial value of the car
k = 0.1     # Depreciation rate (10% per year)
time = 5     # Number of years

Vt = V0 * math.exp(-k * time)
print(f"Value of the car after {time} years: ${Vt:.2f}")

Drug Metabolism 💊

The concentration of a drug in the bloodstream often follows exponential decay as the body metabolizes and eliminates it.

If $C(t)$ is the concentration at time $t$:

C(t) = C_0 * e^{-kt}

• $C_0$ is the initial concentration,
• $k$ is the elimination rate constant.

Electrical Circuits ⚡

In an RC circuit (resistor-capacitor), the voltage across the capacitor decays exponentially when discharging.

V(t) = V_0 * e^{-t/RC}

• $V(t)$ is the voltage at time $t$,
• $V_0$ is the initial voltage,
• $R$ is the resistance,
• $C$ is the capacitance.