The formula to model the decay of a substance using half-life is given by:

$f(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{h}}$

Where:

• $f(t)$ is the amount of substance remaining after time $t$.
• $A_0$ is the initial amount of the substance.
• $t$ is the time in years.
• $h$ is the half-life of the substance.

For plutonium-244, the half-life $h$ is 80,000,000 years. Substituting this value into the formula, we get:

$f(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{80,000,000}}$

This is the function that represents the amount of plutonium-244 remaining as a function of time $t$ (in years).

Final Answer:

$f(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{80,000,000}}$

Would you like to go over how this formula works or have any further questions?

---

Here are 5 related questions for further practice:

1. What is the amount of plutonium-244 remaining after 160,000,000 years if $A_0 = 10$ grams?
2. How does the formula change if the half-life were different?
3. Can you derive the formula for continuous exponential decay using logarithms?
4. What is the decay rate (as a decimal) for plutonium-244?
5. How would you solve for $t$ if the amount of plutonium-244 is reduced to one quarter of its initial amount?

Tip: The half-life formula is a specific case of exponential decay, where the base represents halving after a fixed interval.