Step 1: Isotones in one line.
Isotones are nuclei sharing an equal neutron number \(N=A-Z\) while their proton numbers differ, for example \(_{15}^{31}\text{P}\) and \(_{16}^{32}\text{S}\) (each with 16 neutrons).
Step 2: Build density from a single nucleon.
Since every nucleus packs \(A\) nucleons into a sphere of radius \(R=R_0A^{1/3}\), the volume per nucleon is \(\dfrac{(4/3)\pi R_0^3 A}{A}=\dfrac{4}{3}\pi R_0^3\), independent of \(A\). Density is therefore one nucleon mass divided by that volume:
\(\rho=\dfrac{m}{\frac{4}{3}\pi R_0^3}=\dfrac{3m}{4\pi R_0^3}\).
Step 3: Plug in numbers.
\(\rho=\dfrac{3(1.67\times10^{-27})}{4\pi(1.2\times10^{-15})^3}\). Cube of \(1.2\times10^{-15}\) is \(1.728\times10^{-45}\) m\(^3\).
Step 4: Evaluate.
\(=\dfrac{5.01\times10^{-27}}{2.172\times10^{-44}}=2.31\times10^{17}\) kg/m\(^3\). This huge, constant value shows nuclear matter is extremely dense.
\[\boxed{\rho\approx 2.3\times10^{17}\ \text{kg/m}^3}\]