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 solve the problem of finding the ratio of the radii of two spherical nuclei with mass numbers 216 and 64, we can follow these steps:

### Step 1: Understand the formula for the radius of a nucleus
The radius \( R \) of a nucleus can be expressed in terms of its mass number \( A \) using the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant approximately equal to \( 1.2 \times 10^{-15} \) meters.

### Step 2: Write the expressions for the radii of the two nuclei
Let \( R_1 \) be the radius of the nucleus with mass number \( A_1 = 216 \) and \( R_2 \) be the radius of the nucleus with mass number \( A_2 = 64 \). We can write:
\[ R_1 = R_0 (216)^{1/3} \]
\[ R_2 = R_0 (64)^{1/3} \]

### Step 3: Find the ratio of the radii
To find the ratio \( \frac{R_1}{R_2} \), we can substitute the expressions for \( R_1 \) and \( R_2 \):
\[ \frac{R_1}{R_2} = \frac{R_0 (216)^{1/3}}{R_0 (64)^{1/3}} \]
The \( R_0 \) cancels out:
\[ \frac{R_1}{R_2} = \frac{(216)^{1/3}}{(64)^{1/3}} = \left(\frac{216}{64}\right)^{1/3} \]

### Step 4: Simplify the fraction
Now, we simplify \( \frac{216}{64} \):
\[ \frac{216}{64} = \frac{27}{8} = \left(\frac{3}{2}\right)^3 \]
Thus,
\[ \frac{R_1}{R_2} = \left(\frac{3}{2}\right)^{1} = \frac{3}{2} \]

### Step 5: State the final answer
Therefore, the ratio of the radii \( \frac{R_1}{R_2} \) is:
\[ \frac{R_1}{R_2} = \frac{3}{2} \]

### Final Answer
The ratio \( \frac{R_1}{R_2} \) is \( \frac{3}{2} \).

---