To find the total volume of ice cream in and on top of the cone, we need to calculate the volume of the cone and add it to the volume of the hemisphere.
Step 1: Calculate the Volume of the Hemisphere
We're given that the volume of the hemisphere is 4 cubic inches. The volume \( V \) of a hemisphere is related to its radius \( r \) by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
Given that the volume of the hemisphere is 4 in³:
\[ \frac{2}{3} \pi r^3 = 4 \]
To find \( r^3 \), we rearrange the equation:
\[ r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Step 2: Calculate the Diameter of the Hemisphere
Since the diameter \( D \) is twice the radius, we have:
\[ D = 2r \]
Step 3: Relate Cone Dimensions to Hemisphere
According to the problem, the diameter of the cone is equal to its height. Let's denote \( h \) as the height of the cone. The diameter of the cone \( D \) is:
\[ D = 2r \]
Since the diameter of the cone equals its height \( h \):
\[ h = D = 2r \]
Step 4: Express the Height in Terms of r
So we need \( h \) in terms of \( r \):
Given \( D = 2r \),
\[ h = 2r \]
Step 5: Calculate the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
We also know that \( h = 2r \). Substituting this into the cone volume formula provides:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Step 6: Use the Volume of the Hemisphere to Find Cone Volume
From our earlier work, we know:
\[ \frac{2}{3} \pi r^3 = 4 \quad \Rightarrow \quad V_{\text{cone}} = 4 \]
Step 7: Find Total Volume
Now, we can find the total volume of the ice cream, which is the sum of the volume of the ice cream cone and the volume of the hemisphere:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 + 4 = 8 \text{ in}^3 \]
Conclusion
The total volume of ice cream in and on top of the cone is:
\[ \boxed{8 \text{ in}^3} \]