Applications of Exponential Functions in Continuous Compounding and Radioactive Decay

Exponential Functions and Their Applications

Exponential functions are widely used in calculus to model processes that change at rates proportional to their current value. Two common applications are continuous compounding in finance and radioactive decay in physics.

Continuous Compounding of Interest

When interest is compounded continuously, the amount of money grows according to an exponential function. The formula for continuous compounding is:

• Formula: $A = Pe^{rt}$ where: $A$ = final amount $P$ = initial principal $r$ = annual interest rate (as a decimal) $t$ = time in years $e$ = Euler's number (approximately 2.71828)

• Solving for Time to Reach a Multiple of the Principal: To find the time $t$ required for the principal to grow by a factor $k$ (e.g., double, triple), set $A = kP$ and solve for $t$: $kP = Pe^{rt}$ $k = e^{rt}$ $\ln k = rt$ $t = \dfrac{\ln k}{r}$

• Example: How long will it take money to triple if it is invested at 5.5% compounded continuously? Here, $k = 3$, $r = 0.055$ $t = \dfrac{\ln 3}{0.055}$ $t \approx \dfrac{1.0986}{0.055} \approx 19.97$ years

Radioactive Decay and Exponential Models

Radioactive decay is another process modeled by exponential functions, where the quantity of a substance decreases at a rate proportional to its current amount.

• Decay Formula: $N(t) = N_0 e^{-rt}$ where: $N(t)$ = amount at time $t$ $N_0$ = initial amount $r$ = decay constant (per unit time) $t$ = time

• Half-Life: The half-life is the time required for half of the original substance to decay. Set $N(t) = \dfrac{1}{2}N_0$ and solve for $t$: $\dfrac{1}{2}N_0 = N_0 e^{-rt}$ $\dfrac{1}{2} = e^{-rt}$ $\ln \dfrac{1}{2} = -rt$ $t = \dfrac{\ln(1/2)}{-r} = \dfrac{\ln 2}{r}$

• Example: The continuous compound rate of decay of carbon-14 per year is $r = 0.0001238$. How long will it take a certain amount of carbon-14 to decay to half the original amount? $t = \dfrac{\ln 2}{0.0001238}$ $t \approx \dfrac{0.6931}{0.0001238} \approx 5,598$ years

Summary Table: Exponential Growth and Decay Formulas

Application	General Formula	Time to Reach Fraction/Multiple
Continuous Compounding	$A = Pe^{rt}$	$t = \dfrac{\ln(k)}{r}$, where $k = \dfrac{A}{P}$
Radioactive Decay	$N(t) = N_0 e^{-rt}$	$t = \dfrac{\ln(\text{fraction})}{-r}$ For half-life: $t_{1/2} = \dfrac{\ln 2}{r}$