A scientist receiving a new sample of plutonium knows that the sample will not be of use to him after 9090 percent of the radioactive material in the sample has disintegrated.

If the half-life of this particular plutonium isotope is 113113 days, for how many days after the sample arrives will the scientist be able to make use of the plutonium?

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?

---

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.

A scientist receiving a new sample of plutonium knows that the sample will not be of use to him after 90% of the radioactive material in the sample has disintegrated. If the half-life of this particular plutonium isotope is 113 days, for how many days after the sample arrives will the scientist be able to make use of the plutonium?

Learn how to calculate the time a scientist can use a plutonium sample before 90% of it decays. This problem involves the half-life formula for radioactive decay and exponential decay laws. Suitable for high school students studying physics and algebra, Grades 10-12.

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