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 step by step, we will follow the instructions given in the video transcript and perform the calculations.

### Step 1: Calculate the Radius of the Nucleus
The formula for the radius of the nucleus \( R_n \) in terms of the mass number \( A \) is given by:

\[ R_n = 1.25 \times 10^{-13} \times A^{1/3} \text{ cm} \]

Given that the mass number \( A = 64 \):

\[ R_n = 1.25 \times 10^{-13} \times 64^{1/3} \]

Calculating \( 64^{1/3} \):

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

Now substituting back into the equation:

\[ R_n = 1.25 \times 10^{-13} \times 4 = 5.0 \times 10^{-13} \text{ cm} \]

### Step 2: Calculate the Volume of the Nucleus
The volume \( V_n \) of a sphere is given by the formula:

\[ V_n = \frac{4}{3} \pi R_n^3 \]

Substituting \( R_n = 5.0 \times 10^{-13} \text{ cm} \):

\[ V_n = \frac{4}{3} \pi (5.0 \times 10^{-13})^3 \]

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:

\[ V_n = \frac{4}{3} \pi (125 \times 10^{-39}) = \frac{500}{3} \pi \times 10^{-39} \text{ cm}^3 \]

### Step 3: Calculate the Volume of the Atom
The radius of the atom is given as \( 1 \text{ Å} = 10^{-8} \text{ cm} \).

Now, using the volume formula for the atom \( V_a \):

\[ V_a = \frac{4}{3} \pi R_a^3 \]

Substituting \( R_a = 10^{-8} \text{ cm} \):

\[ V_a = \frac{4}{3} \pi (10^{-8})^3 \]

Calculating \( (10^{-8})^3 \):

\[ (10^{-8})^3 = 10^{-24} \]

Thus,

\[ V_a = \frac{4}{3} \pi (10^{-24}) \text{ cm}^3 \]

### Step 4: Calculate the Fraction of the Atomic Volume Occupied by the Nucleus
The fraction of the atomic volume occupied by the nucleus is given by:

\[ \text{Fraction} = \frac{V_n}{V_a} \]

Substituting the volumes we calculated:

\[ \text{Fraction} = \frac{\frac{500}{3} \pi \times 10^{-39}}{\frac{4}{3} \pi \times 10^{-24}} \]

The \( \pi \) and \( \frac{4}{3} \) cancel out:

\[ \text{Fraction} = \frac{500 \times 10^{-39}}{4 \times 10^{-24}} = \frac{500}{4} \times 10^{-15} = 125 \times 10^{-15} \]

### Step 5: Expressing the Fraction in the Required Form
We need to express the fraction in the form \( x \times 10^{-13} \):

\[ 125 \times 10^{-15} = 1.25 \times 10^{-13} \]

Thus, \( x = 1.25 \).

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

\[ \boxed{1.25} \]