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
Given that the volume of the hemisphere is 4 in³, we can use the formula for the volume of a hemisphere:
\[ V = \frac{2}{3} \pi r^3 \]
Setting this equal to 4 in³:
\[ \frac{2}{3} \pi r^3 = 4 \]
To solve for \( r^3 \), multiply both sides by \(\frac{3}{2}\):
\[ \pi r^3 = 6 \]
Now divide by \(\pi\):
\[ r^3 = \frac{6}{\pi} \]
Step 2: Volume of the Cone
Since diameter of the cone equals its height, let \( h \) be the height of the cone. The radius \( R \) of the cone can be expressed in terms of \( h \) as follows:
\[ R = \frac{d}{2} = \frac{h}{2} \]
The volume \( V \) of a cone is calculated using the formula:
\[ V = \frac{1}{3} \pi R^2 h \]
Substituting \( R = \frac{h}{2} \):
\[ V = \frac{1}{3} \pi \left( \frac{h}{2} \right)^2 h \]
Expanding this:
\[ V = \frac{1}{3} \pi \left( \frac{h^2}{4} \right) h = \frac{1}{12} \pi h^3 \]
Step 3: Relating the Cone to the Hemisphere
Since the volume of the hemisphere is provided to be 4 in³, we use the expression for \( r^3 \) to relate it back to the height of the cone. Notice:
From the volume relationship:
\[ r^3 = \frac{6}{\pi} \]
If the height of the cone \( h \) is expressed in terms of the radius of the hemisphere:
The radius \( r \) of the hemisphere is equivalent to the radius \( R \) of the cone (from the point where it meets the hemisphere), as they are both the same at the top. Thus, we set \( R = r \).
The radius \( R \) can also be expressed in terms of \( h \):
\[ R = \frac{h}{2} \rightarrow r = \frac{h}{2} \]
Thus, we have:
\[ \left( \frac{h}{2} \right)^3 = \frac{6}{\pi} \]
Calculating \( \frac{h^3}{8} = \frac{6}{\pi} \).
Multiplying both sides by \( 8 \):
\[ h^3 = \frac{48}{\pi} \]
Step 4: Volume of the Cone
We can now substitute this value in the cone's volume equation:
\[ V_{cone} = \frac{1}{12} \pi h^3 = \frac{1}{12} \pi \left( \frac{48}{\pi} \right) = \frac{48}{12} = 4 \]
Step 5: Total Volume of Ice Cream
Now, we can find the total volume of ice cream:
\[ V_{total} = V_{cone} + V_{hemisphere} = 4 \text{ in}^3 + 4 \text{ in}^3 = 8 \text{ in}^3 \]
Final Answer
Thus, the total volume of ice cream in and on top of the cone is:
\[ \boxed{8 \text{ in}^3} \]
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