To solve this problem, we need to find the total volume of ice cream in and on top of the cone.
Step 1: Volume of the Hemisphere We are told that the volume of the hemisphere of ice cream on top of the cone is 4 cubic inches. The formula for the volume \( V \) of a hemisphere is:
\[ V = \frac{2}{3} \pi r^3 \]
Given that the volume is 4 in³:
\[ \frac{2}{3} \pi r^3 = 4 \]
To solve for \( r \), we can rearrange this equation:
\[ r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Now we can find \( r \):
\[ r = \left(\frac{6}{\pi}\right)^{1/3} \]
Step 2: Finding the Radius and Height of the Cone Next, we note that the diameter of the cone equals its height. The radius \( R \) of the cone is equal to the radius of the hemisphere \( r \) since the hemisphere sits directly on top of the cone. Therefore, we have:
\[ R = r = \left(\frac{6}{\pi}\right)^{1/3} \]
The diameter \( D \) of the cone is given by:
\[ D = 2R = 2\left(\frac{6}{\pi}\right)^{1/3} \]
The height \( h \) of the cone is also \( D \):
\[ h = R = \left(\frac{6}{\pi}\right)^{1/3} \]
Using these, we can express the volume of the cone using its formula:
\[ V_{cone} = \frac{1}{3} \pi R^2 h \]
Substituting \( R \) and \( h \):
\[ V_{cone} = \frac{1}{3} \pi \left(\left(\frac{6}{\pi}\right)^{1/3}\right)^2 \left(\left(\frac{6}{\pi}\right)^{1/3}\right) \]
Calculating \( R^2 \):
\[ R^2 = \left(\left(\frac{6}{\pi}\right)^{1/3}\right)^2 = \left(\frac{6^2}{\pi^{2}}\right)^{1/3} = \frac{36^{1/3}}{\pi^{2/3}} \]
Now substituting this into the volume of the cone formula:
\[ V_{cone} = \frac{1}{3} \pi \left(\frac{36^{1/3}}{\pi^{2/3}}\right) \left(\frac{6}{\pi}\right)^{1/3} \]
Combining the terms:
\[ = \frac{1}{3} \cdot 6^{1/3} \cdot 36^{1/3} \cdot \frac{\pi}{\pi} = \frac{1}{3} \cdot (6 \cdot 36)^{1/3} = \frac{1}{3} \cdot (216)^{1/3} = \frac{1}{3} \cdot 6 = 2 \text{ in}^3 \]
Step 3: Total Volume of Ice Cream Now we can find the total volume of ice cream by adding the volume in the cone and the volume in the hemisphere:
\[ V_{total} = V_{cone} + V_{hemisphere} = 2 \text{ in}^3 + 4 \text{ in}^3 = 6 \text{ in}^3 \]
Thus, the total volume of ice cream in and on top of the cone is \( \boxed{6} \) cubic inches.