The nucleus of an atom is spherical. The relation between radius of the nucleus and mass number A is given by `1.25xx10^(-13)xxA^((1)/(3))cm`. If radius of atom is one`Å` and the mass number is 64, then the fraction of the atomic volume that is occupied by the nucleus is `(x)xx10^(-13)`. Calculate x

Answer

To solve the problem, we need to find the fraction of the atomic volume that is occupied by the nucleus of an atom given the mass number \( A = 64 \) and the radius of the atom as \( 1 \, \text{Å} \) (which is \( 10^{-8} \, \text{cm} \)).

### Step-by-Step Solution:

1. **Calculate the Radius of the Nucleus:**
The formula for the radius of the nucleus \( R \) in terms of mass number \( A \) is given by:
\[ R = 1.25 \times 10^{-13} \times A^{1/3} \, \text{cm} \]
Substituting \( A = 64 \):
\[ R = 1.25 \times 10^{-13} \times 64^{1/3} \]
First, calculate \( 64^{1/3} \):
\[ 64^{1/3} = 4 \]
Now, substituting this value back:
\[ R = 1.25 \times 10^{-13} \times 4 = 5.0 \times 10^{-13} \, \text{cm} \]

2. **Calculate the Volume of the Nucleus:**
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Substituting the radius of the nucleus:
\[ V_{\text{nucleus}} = \frac{4}{3} \pi (5.0 \times 10^{-13})^3 \]
Calculating \( (5.0 \times 10^{-13})^3 \):
\[ (5.0)^3 = 125 \quad \text{and} \quad (10^{-13})^3 = 10^{-39} \]
Thus,
\[ (5.0 \times 10^{-13})^3 = 125 \times 10^{-39} \, \text{cm}^3 \]
Now, substituting back into the volume formula:
\[ V_{\text{nucleus}} = \frac{4}{3} \pi (125 \times 10^{-39}) = \frac{500\pi}{3} \times 10^{-39} \, \text{cm}^3 \]

3. **Calculate the Volume of the Atom:**
The radius of the atom is given as \( 1 \, \text{Å} = 10^{-8} \, \text{cm} \). Thus,
\[ V_{\text{atom}} = \frac{4}{3} \pi (10^{-8})^3 \]
Calculating \( (10^{-8})^3 \):
\[ (10^{-8})^3 = 10^{-24} \]
Therefore,
\[ V_{\text{atom}} = \frac{4}{3} \pi (10^{-24}) \, \text{cm}^3 \]

4. **Calculate the Fraction of Atomic Volume Occupied by the Nucleus:**
The fraction \( f \) of the atomic volume that is occupied by the nucleus is given by:
\[ f = \frac{V_{\text{nucleus}}}{V_{\text{atom}}} \]
Substituting the volumes we calculated:
\[ f = \frac{\frac{500\pi}{3} \times 10^{-39}}{\frac{4}{3} \pi \times 10^{-24}} = \frac{500 \times 10^{-39}}{4 \times 10^{-24}} \]
Simplifying this expression:
\[ f = \frac{500}{4} \times 10^{-39 + 24} = 125 \times 10^{-15} \]

5. **Express the Fraction in the Required Form:**
We need to express \( f \) in the form \( x \times 10^{-13} \):
\[ f = 1.25 \times 10^{-13} \]
Thus, \( x = 1.25 \).

### Final Answer:
The value of \( x \) is \( 1.25 \).