Answer
\[\underline{7.3\times {{10}^{11}}\text{ g/c}{{\text{m}}^{3}}}\]
Work Step by Step
Convert the radius of the sun into meters as follows:
\[\begin{align}
& r=7.0\times {{10}^{5}}\text{ km}\left( \frac{{{10}^{3}}\text{ m}}{1\text{ km}} \right) \\
& =7.0\times {{10}^{8}}\ \text{cm}
\end{align}\]
Now, the volume of the sun is as follows:
\[V=\frac{4}{3}\pi {{r}^{3}}\]
Substitute 3.14 for \[\pi \] and \[7.0\times {{10}^{10}}\ \text{cm}\] for r:
\[\begin{align}
& V=\frac{4}{3}\left( 3.14 \right){{\left( 7.0\times {{10}^{8}}\ \text{m} \right)}^{3}} \\
& =1.436\times {{10}^{27}}\text{ }{{\text{m}}^{3}}
\end{align}\]
Now, the mass of the sun is calculated as follows:
\[m=\left( d \right)\left( V \right)\]
Here, d is the density and V is the volume.
Substitute \[1.4\times {{10}^{3}}\text{ kg/}{{\text{m}}^{3}}\] for d and \[1.436\times {{10}^{27}}\text{ }{{\text{m}}^{3}}\] for V:
\[\begin{align}
& m=\left( 1.4\times {{10}^{3}}\text{ kg/}{{\text{m}}^{\text{3}}} \right)\left( 1.436\times {{10}^{27}}\text{ }{{\text{m}}^{3}} \right) \\
& =2.01\times {{10}^{30}}\text{ kg}
\end{align}\]
Now, the black hole has a mass of \[1\times {{10}^{3}}\text{ }suns\]. So, the mass of the black hole is calculated as follows:
\[\begin{align}
& {{m}_{\text{b}}}=\left( 2.01\times {{10}^{30}}\text{ kg} \right)\left( \frac{{{10}^{3}}\text{ g}}{1\text{ kg}} \right)\left( 1.0\times {{10}^{3}} \right) \\
& =2.01\times {{10}^{36}}\text{ g}
\end{align}\]
The diameter of the moon is \[2.16\times {{10}^{3}}\text{ }miles\]. Thus, the radius of the moon in centimeters will be as follows:
\[\begin{align}
& {{r}_{\text{m}}}=\left( \frac{2.16\times {{10}^{3}}\text{ miles}}{2} \right)\left( \frac{1.609\text{ km}}{1\text{ mile}} \right)\left( \frac{1000\text{ m}}{1\text{ km}} \right)\left( \frac{100\text{ cm}}{1\text{ m}} \right) \\
& =1.74\times {{10}^{8}}\text{ cm}
\end{align}\]
Now, the radius of the black hole is half the radius of the moon. Thus, the radius of the black hole is as follows:
\[\begin{align}
& {{r}_{\text{b}}}=\frac{1}{2}\left( 1.74\times {{10}^{8}}\text{ cm} \right) \\
& =8.69\times {{10}^{7}}\text{ cm}
\end{align}\]
So, the volume of the black hole is calculated as follows:
\[\begin{align}
& {{V}_{\text{b}}}=\frac{4}{3}\left( 3.14 \right){{\left( 8.69\times {{10}^{7}}\ \text{m} \right)}^{3}} \\
& =2.75\times {{10}^{24}}\text{ c}{{\text{m}}^{3}}
\end{align}\]
Now, the density of the black hole is calculated as follows:
\[{{d}_{\text{b}}}=\frac{{{m}_{\text{b}}}}{{{V}_{\text{b}}}}\]
Substitute \[2.75\times {{10}^{24}}\text{ c}{{\text{m}}^{3}}\] for \[{{V}_{\text{b}}}\] and \[2.01\times {{10}^{36}}\text{ g}\] for \[{{m}_{\text{b}}}\]:
\[\begin{align}
& {{d}_{\text{b}}}=\frac{2.01\times {{10}^{36}}\text{ g}}{2.75\times {{10}^{24}}\text{ c}{{\text{m}}^{3}}} \\
& =7.3\times {{10}^{11}}\text{ g/c}{{\text{m}}^{3}}
\end{align}\]
The density of the black hole is \[\underline{7.3\times {{10}^{11}}\text{ g/c}{{\text{m}}^{\text{3}}}}\].
