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The problem involves:
1. A radioactive decay equation for Plutonium, \( p(t) = p_0 e^{-\gamma t} \), where \( \gamma = 9.2 \times 10^{-10} \), \( t \) is in seconds, \( p_0 \) is the initial radioactivity, and \( p(t) \) represents the radioactivity at time \( t \).
2. You are asked to:
   a) Find the inverse of this function and explain its meaning.
   b) Calculate the time for Plutonium with an initial radioactivity of \( p_0 = 100 \) to decay to 5% of its initial radioactivity.

Let's solve each part step by step:

### Part (a): Finding the inverse of \( p(t) = p_0 e^{-\gamma t} \)
We need to solve for \( t \) in terms of \( p(t) \) and \( p_0 \).

Given:
\[
p(t) = p_0 e^{-\gamma t}
\]
To find the inverse, solve for \( t \):

1. Divide both sides by \( p_0 \):
   \[
   \frac{p(t)}{p_0} = e^{-\gamma t}
   \]
2. Take the natural logarithm (ln) of both sides to remove the exponential:
   \[
   \ln \left( \frac{p(t)}{p_0} \right) = -\gamma t
   \]
3. Solve for \( t \):
   \[
   t = \frac{-\ln \left( \frac{p(t)}{p_0} \right)}{\gamma}
   \]

Thus, the inverse function is:
\[
t = \frac{-\ln \left( \frac{p(t)}{p_0} \right)}{\gamma}
\]

**Meaning**: The inverse function \( t \) tells you the time it takes for the radioactivity to decay to a specified level \( p(t) \), given the initial radioactivity \( p_0 \) and the decay constant \( \gamma \).

### Part (b): Time for Plutonium to decay to 5% of its initial radioactivity

Here, you're given that \( p_0 = 100 \) and need to find the time \( t \) when \( p(t) = 0.05 \times p_0 \). This means:
\[
p(t) = 5
\]
Using the inverse function from part (a):
\[
t = \frac{-\ln \left( \frac{p(t)}{p_0} \right)}{\gamma}
\]
Substitute \( p(t) = 5 \), \( p_0 = 100 \), and \( \gamma = 9.2 \times 10^{-10} \):

\[
t = \frac{-\ln \left( \frac{5}{100} \right)}{9.2 \times 10^{-10}}
\]
\[
t = \frac{-\ln (0.05)}{9.2 \times 10^{-10}}
\]
Now, let's calculate that value.

I'll perform this calculation.The time it takes for Plutonium to decay to 5% of its initial radioactivity is approximately \( 3.26 \times 10^9 \) seconds, or about 103.4 years.

Would you like any further details or clarification on these calculations?

Here are five related questions you can explore:
1. What is the half-life of Plutonium based on the given decay constant?
2. How does the decay constant \( \gamma \) affect the rate of decay?
3. What are the units of \( \gamma \) and how do they impact the equation?
4. What is the radioactivity after 50 years for this Plutonium sample?
5. Can this model be applied to other radioactive elements? How would their decay constants change?

**Tip**: Always check the units of each variable to ensure the results are consistent, especially in exponential decay problems.

Plutonium is a radioactive atom, which decays at a rate of γ = (9.2) × 10^−10. Let p(t) denote the radioactivity at time t (t is in seconds) and p₀ the initial radioactivity. a) Find the inverse of this function and explain its meaning. b) How many years does it take for a piece of plutonium of initial radioactivity p₀ = 100 to decay to 5% of its initial radioactivity?

Learn how to solve a problem involving the radioactive decay of plutonium with a decay constant γ = 9.2 × 10^−10. This question explores finding the inverse of the exponential decay function and calculating the time it takes for plutonium to decay to 5% of its initial radioactivity.

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