Let M(t) be the mass of a radioactive substance at year t, which gradually disintegrates. 1) M(t) satisfies natural exponential decay 2) decreases at a fixed instantaneous rate of k 3)rate of disintegration is proportional to the amount remaining at time t. 4)After 15 years, it is determined that 0.043% of the initial amount is disintegrated. Find the half life.

To solve the problem of radioactive decay, we start with the model for natural exponential decay given by the equation:

\[
M(t) = M_0 e^{-kt}
\]

where:
- \( M(t) \) is the mass at time \( t \),
- \( M_0 \) is the initial mass,
- \( k \) is the decay constant,
- \( t \) is time in years.

### Step 1: Finding the decay constant \( k \)

We know that after 15 years, 0.043% of the initial amount has disintegrated. This means that 99.957% remains:

\[
M(15) = M_0 (1 - 0.00043) = M_0 \cdot 0.99957
\]

Substituting this into the decay equation, we have:

\[
M_0 \cdot 0.99957 = M_0 e^{-15k}
\]

Dividing both sides by \( M_0 \) (assuming \( M_0 \neq 0 \)) gives:

\[
0.99957 = e^{-15k}
\]

### Step 2: Solving for \( k \)

Taking the natural logarithm of both sides:

\[
\ln(0.99957) = -15k
\]

Now, we can solve for \( k \):

\[
k = -\frac{\ln(0.99957)}{15}
\]

Calculating this:

\[
k \approx -\frac{-0.00043}{15} \approx 0.00002867 \text{ (rounded to 8 significant figures)}
\]

### Step 3: Finding the half-life \( T_{1/2} \)

The half-life is given by the formula:

\[
T_{1/2} = \frac{\ln(2)}{k}
\]

Substituting our value of \( k \):

\[
T_{1/2} = \frac{\ln(2)}{0.00002867}
\]

Calculating this gives:

\[
T_{1/2} \approx \frac{0.693147}{0.00002867} \approx 24262.2 \text{ years}
\]

### Final Result

Thus, the half-life of the radioactive substance is approximately **24,262 years**.

### Additional Information

If you have any questions or would like to see further details on any part of this solution, let me know! Here are some related questions you might consider:

1. What are the implications of a long half-life on the stability of a substance?
2. How does the decay constant \( k \) relate to the half-life?
3. Can you explain the process of exponential decay in more detail?
4. How would this model change if the decay was not proportional to the amount remaining?
5. What real-world applications use the concept of half-life in their calculations?

**Tip:** To better understand decay processes, it can be helpful to graph the decay function and observe how quickly the quantity decreases over time.

Let M(t) be the mass of a radioactive substance at year t, which gradually disintegrates. After 15 years, it is determined that 0.043% of the initial amount is disintegrated. Find the half-life.

Explore how to calculate the half-life of a radioactive substance based on its decay over 15 years, where only 0.043% remains. This guide details the exponential decay model and provides step-by-step calculations suitable for high school students.

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