Given that the volume of the hemisphere of ice cream is \( 4 , \text{in}^3 \), we can use the formula for the volume of a hemisphere with radius \( r \):
\[ V = \frac{2}{3}\pi r^3 \]
Setting this equal to the volume given:
\[ \frac{2}{3}\pi r^3 = 4 \]
To solve for \( r^3 \), we first multiply both sides by \( \frac{3}{2} \):
\[ \pi r^3 = 6 \]
Now, divide both sides by \( \pi \):
\[ r^3 = \frac{6}{\pi} \]
Next, we take the cube root to find \( r \):
\[ r = \left(\frac{6}{\pi}\right)^{\frac{1}{3}} \]
Next, we need to find the total volume of ice cream in and on top of the cone. The cone's height \( h \) is equal to its diameter, which means:
\[ h = 2r \]
Since the diameter of the cone is equal to its height, its radius \( R \) is half of that:
\[ R = \frac{h}{2} = \frac{2r}{2} = r \]
Now we can calculate the volume of the cone using its volume formula \( V = \frac{1}{3}\pi R^2 h \):
We substitute \( R \) and \( h \):
\[ V_{\text{cone}} = \frac{1}{3}\pi r^2 (2r) = \frac{2}{3}\pi r^3 \]
Since we already have \( r^3 = \frac{6}{\pi} \), we can substitute that into the equation:
\[ V_{\text{cone}} = \frac{2}{3}\pi \left(\frac{6}{\pi}\right) = \frac{2}{3} \times 6 = 4 , \text{in}^3 \]
Now we can find the total volume of ice cream in and on top of the cone by adding the volume of the cone and the volume of the hemisphere:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]
Thus, the total volume of ice cream in and on top of the cone is:
\[ \boxed{8 , \text{in}^3} \]