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 then add the volume of the hemisphere on top.
Step 1: Volume of the Hemisphere
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
where \( r \) is the radius of the hemisphere. From the problem, we know that the volume of the hemisphere is \( 4 , \text{in}^3 \). Therefore, we can set up the equation:
\[ \frac{2}{3} \pi r^3 = 4 \]
To solve for \( r^3 \), we first multiply both sides by \( \frac{3}{2} \):
\[ \pi r^3 = 6 \]
Next, we divide both sides by \( \pi \):
\[ r^3 = \frac{6}{\pi} \]
Finally, we take the cube root to find \( r \):
\[ r = \left(\frac{6}{\pi}\right)^{1/3} \]
Step 2: Volume of the Cone
We know that the diameter of the cone is equal to its height. Denote the height of the cone as \( h \) and the radius \( R \) of the cone's base as \( R = \frac{D}{2} \). Since the diameter \( D \) is equal to the height \( h \), we have \( R = \frac{h}{2} \).
Now, the volume \( V_{cone} \) of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi R^2 h \]
Substituting \( R \) in terms of \( h \):
\[ V_{cone} = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \]
Calculating this gives:
\[ V_{cone} = \frac{1}{3} \pi \cdot \frac{h^2}{4} \cdot h = \frac{1}{12} \pi h^3 \]
Step 3: Relating \( h \) to the Hemisphere Radius
We have \( r \) from the hemisphere, which is equal to \( \frac{h}{2} \) because the cone's diameter equals its height. Thus,
\[ r = \frac{h}{2} \]
From our earlier equation for \( r \), we can express \( h \) in terms of \( r \):
\[ h = 2r \]
Substituting \( h = 2r \) into the cone's volume equation gives:
\[ V_{cone} = \frac{1}{12} \pi (2r)^3 = \frac{1}{12} \pi \cdot 8r^3 = \frac{2}{3} \pi r^3 \]
Step 4: Total Volume Calculation
Now, we add the volumes of the cone and hemisphere:
The volume of the hemisphere is:
\[ V_{hemisphere} = 4 \]
And the volume of the cone is:
\[ V_{cone} = \frac{2}{3} \pi r^3 \]
Substituting \( r^3 = \frac{6}{\pi} \) into the cone volume:
\[ V_{cone} = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2}{3} \cdot 6 = 4 \]
Final Step: Total Volume of Ice Cream
Adding both volumes together:
\[ V_{total} = V_{cone} + V_{hemisphere} = 4 + 4 = 8 , \text{in}^3 \]
Conclusion: The total volume of ice cream in and on top of the cone is \( \boxed{8} , \text{in}^3 \).