The amount of plutonium-244 remaining as a function of time can be expressed using the exponential decay formula:

$A(t) = A_0 \cdot e^{-kt}$

Where:

• $A(t)$ is the amount remaining at time $t$.
• $A_0$ is the initial amount.
• $k$ is the decay constant.
• $t$ is the time in years.

First, we need to find the decay constant $k$. The relationship between the half-life and the decay constant is given by:

$k = \frac{\ln(2)}{\text{half-life}}$

Substituting the half-life of plutonium-244, which is 80,000,000 years, we get:

$k = \frac{\ln(2)}{80000000}$

Now, substituting this value of $k$ into the exponential decay formula:

$A(t) = A_0 \cdot e^{-\frac{\ln(2)}{80000000} \cdot t}$

This is the exact expression for the amount of plutonium-244 remaining as a function of time.

Would you like further explanation, or do you have any questions?

Related Questions:

1. How is the decay constant $k$ derived from the half-life equation?
2. Can you explain how to compute $e^{-kt}$ for a given $t$?
3. How does the half-life affect the decay rate over time?
4. How much plutonium-244 remains after 160,000,000 years?
5. What is the general form of an exponential decay function in real-world scenarios?

Tip: Always check if the units of time match the half-life when applying the formula to specific problems.