Two spherical nuclei have mass number 216 and 64 with their radii `R_(1) and R_(2)` respectively. The ratio, `(R_(1))/(R_(2))` is equal to

Options

A. `3:2`
B. `1:3`
C. `1:2`
D. `2:3`

Explanation

To find the ratio of the radii \( R_1 \) and \( R_2 \) of two spherical nuclei with mass numbers 216 and 64, we can follow these steps:

### Step 1: Understand the formula for nuclear radius
The radius \( R \) of a spherical nucleus can be expressed using the formula:
\[ R = R_0 A^{1/3} \]
where:
- \( R_0 \) is a constant (approximately \( 1.25 \times 10^{-10} \) m),
- \( A \) is the mass number of the nucleus.

### Step 2: Write the expressions for the radii
For the two nuclei, we can express their radii as:
\[ R_1 = R_0 A_1^{1/3} \]
\[ R_2 = R_0 A_2^{1/3} \]
where \( A_1 = 216 \) and \( A_2 = 64 \).

### Step 3: Set up the ratio of the radii
To find the ratio \( \frac{R_1}{R_2} \), we can substitute the expressions for \( R_1 \) and \( R_2 \):
\[ \frac{R_1}{R_2} = \frac{R_0 A_1^{1/3}}{R_0 A_2^{1/3}} \]

### Step 4: Simplify the ratio
The \( R_0 \) terms cancel out:
\[ \frac{R_1}{R_2} = \frac{A_1^{1/3}}{A_2^{1/3}} = \left(\frac{A_1}{A_2}\right)^{1/3} \]

### Step 5: Substitute the mass numbers
Now, substitute \( A_1 = 216 \) and \( A_2 = 64 \):
\[ \frac{R_1}{R_2} = \left(\frac{216}{64}\right)^{1/3} \]

### Step 6: Calculate the ratio
Calculate \( \frac{216}{64} \):
\[ \frac{216}{64} = \frac{27}{8} = 3.375 \]
Now take the cube root:
\[ \frac{R_1}{R_2} = (3.375)^{1/3} = \frac{3}{2} \]

### Final Answer
Thus, the ratio \( \frac{R_1}{R_2} \) is:
\[ \frac{R_1}{R_2} = \frac{3}{2} \]