This is similar to Try It #1 in the OpenStax text.
The half-life of plutonium-244 is 80,000,000 years. Express the amount of plutonium-244 remaining as a function of time, measured in years with starting quantity A0
.

Enter the exact answer.

Do not enter any commas in your answer.

 f(t) = 

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?

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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.

The half-life of plutonium-244 is 80,000,000 years. Express the amount of plutonium-244 remaining as a function of time, measured in years with starting quantity A0. Enter the exact answer.

Explore how to express the amount of plutonium-244 remaining as a function of time, using its half-life of 80,000,000 years. Learn how exponential decay works and apply the formula f(t) = A0 * (1/2)^(t/80,000,000) to calculate the remaining amount of plutonium over time.

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