Answer
See the detailed answer below.
Work Step by Step
To find this percentage, we need to find the atom's volume and the nucleus's volume and then divide one over the other.
$$\dfrac{{V_{\rm nucleus}}}{V_{\rm atom}}\times 100\%=\dfrac{\frac{4\pi}{3}r_{\rm nucleus}^3}{\frac{4\pi}{3}r_{\rm atom}^3}\times 100\%=\dfrac{r^3_{\rm nucleus}}{r^3_{\rm atom}}\times 100\%$$
Plug the known;
$$\dfrac{{V_{\rm nucleus}}}{V_{\rm atom}}\times 100\% =\dfrac{(1\times 10^{-14})^3}{(1.2\times 10^{-10})^3}\times 100\%=\bf 5.79\times 10^{-11}\%$$
Therefore, the percentage of the space inside the atom is given by
$$V_{\rm space}=100\%- 5.79\times 10^{-11}\%=\color{red}{\bf 99.99999999421296}\%$$
$$V_{\rm mass}=\color{red}{ \bf 0.000000000058}\%$$
