The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is `(1000)/x` , where x is _____.

Answer

To solve the problem, we will use the relationship between the radius of a nucleus and its mass number. The radius \( R \) of a nucleus can be expressed in terms of its mass number \( A \) as:

\[ R = R_0 A^{1/3} \]

where \( R_0 \) is a constant (approximately \( 1.2 \) fermi).

### Step 1: Set up the relationship for the first nucleus
For the first nucleus with mass number \( A_1 = 64 \) and radius \( R_1 = 4.8 \) fermi, we can write:

\[ R_1 = R_0 A_1^{1/3} \]

Substituting the values we have:

\[ 4.8 = R_0 (64)^{1/3} \]

### Step 2: Calculate \( R_0 \)
To find \( R_0 \), we first calculate \( (64)^{1/3} \):

\[ (64)^{1/3} = 4 \]

Now substituting back into the equation:

\[ 4.8 = R_0 \cdot 4 \]

Solving for \( R_0 \):

\[ R_0 = \frac{4.8}{4} = 1.2 \text{ fermi} \]

### Step 3: Set up the relationship for the second nucleus
For the second nucleus with radius \( R_2 = 4 \) fermi and mass number \( A_2 \), we can write:

\[ R_2 = R_0 A_2^{1/3} \]

Substituting the known values:

\[ 4 = 1.2 A_2^{1/3} \]

### Step 4: Solve for \( A_2 \)
Now we can solve for \( A_2^{1/3} \):

\[ A_2^{1/3} = \frac{4}{1.2} = \frac{40}{12} = \frac{10}{3} \]

Now cubing both sides to find \( A_2 \):

\[ A_2 = \left(\frac{10}{3}\right)^3 = \frac{1000}{27} \]

### Step 5: Relate \( A_2 \) to the given expression
According to the problem, we have:

\[ A_2 = \frac{1000}{x} \]

Setting the two expressions for \( A_2 \) equal to each other:

\[ \frac{1000}{27} = \frac{1000}{x} \]

### Step 6: Solve for \( x \)
Cross-multiplying gives:

\[ 1000x = 1000 \cdot 27 \]

Dividing both sides by 1000:

\[ x = 27 \]

### Final Answer
Thus, the value of \( x \) is:

\[ \boxed{27} \]