If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius?

$ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $
1. The rotational inertia (moment of inertia) of a sphere about an axis through its center is given by:
   \[
   I = \frac{2}{5} m r^2
   \]
   where \( m \) is the mass of the sphere and \( r \) is its radius.

2. Given that the density of the sphere is constant, we can express the density \( \rho \) as:
   \[
   \rho = \frac{m}{\frac{4}{3} \pi r^3}
   \]
   Since the density is constant, we have:
   \[
   \frac{m}{r^3} = \text{constant} \implies m \propto r^3
   \]

3. Let the original sphere have mass \( m_1 \) and radius \( r_1 \). For the new sphere with twice the radius, let the mass be \( m_2 \) and the radius be \( r_2 = 2r_1 \).

4. Since \( m \propto r^3 \), we have:
   \[
   m_2 = \rho \cdot \frac{4}{3} \pi (2r_1)^3 = \rho \cdot \frac{4}{3} \pi \cdot 8r_1^3 = 8 \cdot \left( \rho \cdot \frac{4}{3} \pi r_1^3 \right) = 8m_1
   \]

5. The moment of inertia for the new sphere is:
   \[
   I_2 = \frac{2}{5} m_2 r_2^2
   \]
   Substituting \( m_2 = 8m_1 \) and \( r_2 = 2r_1 \), we get:
   \[
   I_2 = \frac{2}{5} (8m_1) (2r_1)^2 = \frac{2}{5} \cdot 8m_1 \cdot 4r_1^2 = \frac{2}{5} \cdot 32m_1 r_1^2 = 32 \left( \frac{2}{5} m_1 r_1^2 \right) = 32I
   \]

6. Therefore, the rotational inertia of the new sphere is:
   \[
   \boxed{32I}
   \]