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The problem is about finding the half-life of a radioactive material, given that it loses one-third of its mass in 2 minutes. This is a typical problem of exponential decay, and the half-life is the time required for the material to reduce to half of its original mass.

### Step-by-Step Approach:

The formula for exponential decay is:

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

Where:
- \( M(t) \) is the mass at time \( t \),
- \( M_0 \) is the initial mass,
- \( k \) is the decay constant, and
- \( t \) is time.

We know that after 2 minutes, one-third of the mass has decayed. Therefore, the remaining mass is two-thirds of the original mass. So,

\[
M(2) = \frac{2}{3} M_0
\]

Substitute this into the decay equation:

\[
\frac{2}{3} M_0 = M_0 e^{-2k}
\]

Now, cancel \( M_0 \) from both sides:

\[
\frac{2}{3} = e^{-2k}
\]

Take the natural logarithm on both sides:

\[
\ln\left(\frac{2}{3}\right) = -2k
\]

Now solve for \( k \):

\[
k = -\frac{1}{2} \ln\left(\frac{2}{3}\right)
\]

Next, to find the half-life \( t_{1/2} \), use the fact that when \( M(t) = \frac{1}{2} M_0 \):

\[
\frac{1}{2} M_0 = M_0 e^{-kt_{1/2}}
\]

Cancel \( M_0 \):

\[
\frac{1}{2} = e^{-kt_{1/2}}
\]

Take the natural logarithm on both sides:

\[
\ln\left(\frac{1}{2}\right) = -kt_{1/2}
\]

Solve for \( t_{1/2} \):

\[
t_{1/2} = \frac{\ln\left(\frac{1}{2}\right)}{-k}
\]

Now substitute the value of \( k \) found earlier to get the half-life. Let me calculate it for you.The half-life of the radioactive material is approximately 3.42 minutes.

Would you like further clarification on any step? Here are five related questions to help you explore this topic further:

1. How does the decay constant \( k \) relate to the half-life in general?
2. What happens to the half-life if the decay rate increases?
3. How would the equation change if the material lost 50% of its mass in 2 minutes instead?
4. Can exponential decay be applied to non-radioactive processes? If so, how?
5. How does the exponential decay graph behave as time approaches infinity?

**Tip:** The half-life is independent of the initial quantity of the material; it only depends on the decay rate.

If a radioactive material loses one-third of its mass in 2 minutes, determine its half-life.

Learn how to determine the half-life of a radioactive material that loses one-third of its mass in 2 minutes. This problem involves exponential decay and logarithms, suitable for students in Grades 11-12.

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