We are tasked with determining how many days the scientist can use the plutonium, given that its half-life is 113 days and it becomes unusable after 90% of the radioactive material has disintegrated.

Step 1: Radioactive Decay Formula

The general formula for radioactive decay is:

$N(t) = N_0 \cdot e^{-\lambda t}$

Where:

• $N(t)$ is the remaining amount of substance at time $t$.
• $N_0$ is the initial amount of substance.
• $\lambda$ is the decay constant.
• $t$ is the time in days.

Step 2: Decay Constant ($\lambda$)

The decay constant $\lambda$ is related to the half-life $T_{\frac{1}{2}}$ by the formula:

$\lambda = \frac{\ln(2)}{T_{\frac{1}{2}}}$

For this problem, the half-life is $T_{\frac{1}{2}} = 113$ days. Thus, the decay constant is:

$\lambda = \frac{\ln(2)}{113} \approx 0.006133$

Step 3: Fraction of Remaining Plutonium

The scientist can no longer use the plutonium after 90% of it has decayed, meaning that only 10% of the original amount remains. Thus, we set:

$N(t) = 0.1 \cdot N_0$

Substituting this into the decay formula:

$0.1 \cdot N_0 = N_0 \cdot e^{-\lambda t}$

Canceling $N_0$ from both sides:

$0.1 = e^{-\lambda t}$

Step 4: Solve for $t$

To solve for $t$, take the natural logarithm of both sides:

$\ln(0.1) = -\lambda t$

Thus,

$t = \frac{\ln(0.1)}{-\lambda}$

Substituting $\lambda \approx 0.006133$:

$t = \frac{\ln(0.1)}{-0.006133} \approx \frac{-2.302585}{-0.006133} \approx 375.46 \text{ days}$

Final Answer:

The scientist will be able to use the plutonium for approximately 375 days before 90% of it disintegrates.

Would you like further details on any of these steps or have any questions?

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Here are 5 questions to deepen your understanding:

1. How does the decay constant relate to the half-life of a substance?
2. What happens to the remaining quantity of a substance after multiple half-lives?
3. Why is the natural logarithm used in radioactive decay problems?
4. What would the usable time be if the half-life were different?
5. How does the percentage of disintegration affect the time a sample can be used?

Tip: Always double-check if you're using natural logarithms (base $e$) or common logarithms (base 10) in decay-related calculations.