If the nuclear radius of `.13Al^27 ` is `3.6` Fermi, the approximate nuclear radius of `.29Cu^64` in Fermi is :

Options

A. 4.8
B. 3.6
C. 2.4
D. 1.2

Explanation

To determine the approximate nuclear radius of \( ^{64}Cu \) given the nuclear radius of \( ^{27}Al \), we can use the formula for the nuclear radius, which is given by:

\[ r = r_0 A^{1/3} \]

where:
- \( r \) is the nuclear radius,
- \( r_0 \) is a constant (approximately \( 1.2 \) to \( 1.3 \) Fermi),
- \( A \) is the mass number of the nucleus.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- The nuclear radius of \( ^{27}Al \) is \( 3.6 \) Fermi.
- The mass number \( A \) for aluminum (\( ^{27}Al \)) is \( 27 \).
- The mass number \( A \) for copper (\( ^{64}Cu \)) is \( 64 \).

2. **Set Up the Ratio of Radii:**
Using the relationship for nuclear radii, we can set up the following ratio:
\[ \frac{r_{Al}}{r_{Cu}} = \left( \frac{A_{Al}}{A_{Cu}} \right)^{1/3} \]

3. **Substitute the Known Values:**
- Substitute \( r_{Al} = 3.6 \) Fermi, \( A_{Al} = 27 \), and \( A_{Cu} = 64 \):
\[ \frac{3.6}{r_{Cu}} = \left( \frac{27}{64} \right)^{1/3} \]

4. **Calculate the Right Side:**
- Calculate \( \frac{27}{64} \):
\[ \frac{27}{64} = 0.421875 \]
- Now find the cube root:
\[ \left( 0.421875 \right)^{1/3} \approx 0.75 \]

5. **Set Up the Equation:**
Now we can rewrite the equation:
\[ \frac{3.6}{r_{Cu}} = 0.75 \]

6. **Solve for \( r_{Cu} \):**
- Rearranging gives:
\[ r_{Cu} = \frac{3.6}{0.75} \]

7. **Calculate \( r_{Cu} \):**
\[ r_{Cu} = 4.8 \text{ Fermi} \]

### Final Answer:
The approximate nuclear radius of \( ^{64}Cu \) is \( 4.8 \) Fermi.

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