`r_(1)` and `r_(2)` are the radii of atomic nuclei of mass numbers `64` and `27` repsectively. The ratio `(r_(1)//r_(2))` is

Options

A. `64//27`
B. `27//64`
C. `4//3`
D. `1`

Explanation

To find the ratio \( \frac{r_1}{r_2} \) of the radii of atomic nuclei with mass numbers 64 and 27, we can use the formula that relates the radius of a nucleus to its mass number. The radius \( r \) of a nucleus is given by the formula:

\[ r \propto A^{\frac{1}{3}} \]

where \( A \) is the mass number of the nucleus.

### Step 1: Write the relationship for the radii
Given that \( r_1 \) is the radius of the nucleus with mass number \( A_1 = 64 \) and \( r_2 \) is the radius of the nucleus with mass number \( A_2 = 27 \), we can express the ratio of the radii as:

\[ \frac{r_1}{r_2} = \left(\frac{A_1}{A_2}\right)^{\frac{1}{3}} \]

### Step 2: Substitute the mass numbers
Now, substituting the values of \( A_1 \) and \( A_2 \):

\[ \frac{r_1}{r_2} = \left(\frac{64}{27}\right)^{\frac{1}{3}} \]

### Step 3: Simplify the fraction
Next, we simplify the fraction \( \frac{64}{27} \):

\[ \frac{64}{27} = \left(\frac{4}{3}\right)^3 \]

### Step 4: Take the cube root
Now we take the cube root:

\[ \frac{r_1}{r_2} = \left(\left(\frac{4}{3}\right)^3\right)^{\frac{1}{3}} = \frac{4}{3} \]

### Conclusion
Thus, the ratio of the radii of the atomic nuclei is:

\[ \frac{r_1}{r_2} = \frac{4}{3} \]