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

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Enter the exact answer.

 

Enclose arguments of functions in parentheses and include a multiplication sign between terms. For example, c*log(h)

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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.

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.

Understand how to express the remaining amount of plutonium-244 using the exponential decay formula, based on a half-life of 80,000,000 years. This problem introduces the concept of decay constants and shows the relationship between half-life and logarithms, suitable for students in Grades 11-12.

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