Let's break down the problem into two parts:

Part (a): Finding the Inverse of the Function

The decay equation given is:

$p(t) = p_0 e^{-rt}$

To find the inverse, we want to express time $t$ as a function of $p(t)$. Here's how to do it:

1. Start with the given equation:

   $p(t) = p_0 e^{-rt}$

2. Isolate the exponential term by dividing both sides by $p_0$:

   $\frac{p(t)}{p_0} = e^{-rt}$

3. Take the natural logarithm (ln) of both sides to remove the exponential:

   $\ln\left(\frac{p(t)}{p_0}\right) = \ln\left(e^{-rt}\right)$

   Simplifying the right side:

   $\ln\left(\frac{p(t)}{p_0}\right) = -rt$

4. Solve for $t$ by dividing both sides by $-r$:

   $t = -\frac{1}{r} \ln\left(\frac{p(t)}{p_0}\right)$

So, the inverse function is:

$t(p) = -\frac{1}{r} \ln\left(\frac{p}{p_0}\right)$

Meaning of the Inverse Function:

The inverse function $t(p) = -\frac{1}{r} \ln\left(\frac{p}{p_0}\right)$ gives the time $t$ it takes for the radioactivity to decay from its initial value $p_0$ to a specific value $p$. Essentially, it tells us how long it will take for the radioactivity to decrease to a certain level.

Part (b): Time for Plutonium to Decay to 5% of its Initial Radioactivity

We are asked to find the time $t$ it takes for the radioactivity to decay to 5% of its initial value. The initial radioactivity $p_0 = 100$, and we want $p(t) = 0.05 \times p_0$.

Using the inverse function derived in part (a):

$t = -\frac{1}{r} \ln\left(\frac{p(t)}{p_0}\right)$

Substitute the given values:

$t = -\frac{1}{(9.2 \times 10^{-13})} \ln\left(\frac{0.05 \times p_0}{p_0}\right)$

Simplify the fraction inside the logarithm:

$t = -\frac{1}{(9.2 \times 10^{-13})} \ln(0.05)$

Calculate the logarithm and the time:

$\ln(0.05) \approx -2.9957$

$t \approx -\frac{1}{9.2 \times 10^{-13}} \times (-2.9957)$

$t \approx \frac{2.9957}{9.2 \times 10^{-13}} \approx 3.26 \times 10^{12} \text{ seconds}$

Convert seconds to years:

$t \approx \frac{3.26 \times 10^{12}}{60 \times 60 \times 24 \times 365} \approx 103,400 \text{ years}$

So, it takes approximately 103,400 years for the radioactivity of the plutonium to decay to 5% of its initial value.

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Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How does the decay constant $r$ affect the time it takes for a substance to decay to a certain percentage of its initial value?
2. What is the half-life of plutonium given the decay rate $r = 9.2 \times 10^{-13}$?
3. How would the time change if the decay constant were different?
4. How is the natural logarithm used in other exponential decay processes, such as in population studies or finance?
5. What is the relationship between half-life and decay constant $r$ in radioactive decay?

Tip: When working with exponential decay, always check if you need to convert time units (e.g., seconds to years) to make your answer more interpretable!