The nuclear radius of `Pb^(208)` is is 8.874 fm. What will be the nuclear radius of `Ca^(44)` ?

Answer

To find the nuclear radius of \( \text{Ca}^{44} \) given the nuclear radius of \( \text{Pb}^{208} \), we can use the relationship between the nuclear radius and the mass number. The nuclear radius \( R \) is directly proportional to the cube root of the mass number \( A \):

\[ R \propto A^{1/3} \]

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Nuclear radius of \( \text{Pb}^{208} \) (denoted as \( R_1 \)) = 8.874 fm
- Mass number of \( \text{Pb}^{208} \) (denoted as \( A_1 \)) = 208
- Mass number of \( \text{Ca}^{44} \) (denoted as \( A_2 \)) = 44

2. **Set Up the Relationship:**
According to the relationship, we can express the ratio of the nuclear radii as follows:
\[ \frac{R_1}{R_2} = \left( \frac{A_1}{A_2} \right)^{1/3} \]
Where \( R_2 \) is the nuclear radius of \( \text{Ca}^{44} \).

3. **Substituting the Known Values:**
Substitute \( R_1 \), \( A_1 \), and \( A_2 \) into the equation:
\[ \frac{8.874}{R_2} = \left( \frac{208}{44} \right)^{1/3} \]

4. **Calculate the Ratio of Mass Numbers:**
Calculate \( \frac{208}{44} \):
\[ \frac{208}{44} = 4.727 \]

5. **Calculate the Cube Root:**
Now, calculate the cube root:
\[ \left( 4.727 \right)^{1/3} \approx 1.669 \]

6. **Rearranging to Solve for \( R_2 \):**
Rearranging the equation to solve for \( R_2 \):
\[ R_2 = \frac{8.874}{1.669} \]

7. **Final Calculation:**
Performing the division:
\[ R_2 \approx 5.296 \text{ fm} \]

### Final Answer:
The nuclear radius of \( \text{Ca}^{44} \) is approximately **5.296 fm**.