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The problem involves exponential decay, where the rate of decomposition of radium is proportional to the amount present. The exponential decay formula is given by:

\[
N(t) = N_0 e^{-kt}
\]

Where:
- \( N(t) \) is the amount remaining after time \( t \).
- \( N_0 \) is the initial amount.
- \( k \) is the decay constant.
- \( t \) is the time.

From the problem, it is given that 1.1% decomposes in 25 years. This means 98.9% remains after 25 years, so we can use this to find the decay constant \( k \).

### Step 1: Find the decay constant \( k \)
We know:
\[
\frac{N(25)}{N_0} = 0.989
\]

Substitute into the exponential decay equation:

\[
0.989 = e^{-25k}
\]

Taking the natural logarithm of both sides:

\[
\ln(0.989) = -25k
\]

Solve for \( k \):

\[
k = -\frac{\ln(0.989)}{25}
\]

### Step 2: Solve for half-life (a)
To find the half-life, we want \( N(t) = \frac{N_0}{2} \). This gives:

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

Take the natural logarithm of both sides:

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

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

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

### Step 3: Solve for 20% decomposition (b)
We want 20% decomposed, which means 80% remains:

\[
0.80 = e^{-kt}
\]

Taking the natural logarithm of both sides:

\[
\ln(0.80) = -kt
\]

Solve for \( t \):

\[
t = \frac{\ln(0.80)}{-k}
\]

### Step 4: Solve for the percentage decomposed in 50 years (c)
We want to find the percentage decomposed after 50 years:

\[
\frac{N(50)}{N_0} = e^{-50k}
\]

We can then calculate the percentage remaining and subtract it from 100% to get the decomposed percentage.

Let me compute the value of \( k \) and use it to solve these parts.### Results:
1. **Decay constant**: \( k = 0.000442 \, \text{per year} \).
   
2. **Half-life**:
   It will take approximately **1566.65 years** for half of the original radium amount to decompose.

3. **Time for 20% of the original amount to decompose**:
   It will take approximately **504.35 years** for 20% of the original amount to decompose (i.e., 80% remains).

4. **Percentage decomposed in 50 years**:
   After 50 years, approximately **2.19%** of the original radium amount will have decomposed.

---

Do you need more details or clarifications?  
Here are five related questions for further exploration:
1. What would happen to the decomposition rate if the decay constant \( k \) were larger?
2. How does the half-life relate to different percentages of the substance remaining?
3. How is the decomposition time affected if only 5% decomposes in the first 25 years?
4. What is the formula for calculating the time it takes for a specific percentage of decomposition?
5. How can exponential decay be applied in other real-world scenarios like carbon dating?

**Tip**: Exponential decay problems are common in physics, biology, and finance—mastering this concept helps in various scientific fields!

Radium decomposes at the rate proportional to the amount present. It is found out that in 25 years, 1.1% of a certain amount decomposed. Determine approximately how long it will take for: a) One half of the original amount to decompose. b) 20% of the original amount to decompose. c) What % will decompose in 50 years.

Solve the radium decomposition problem using exponential decay formulas. Find the half-life of radium, the time for 20% decomposition, and the percentage decomposed in 50 years. This is suitable for advanced high school and undergraduate students.

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