To find the total volume of ice cream in and on top of the cone, we need to calculate the volume of both the cone and the hemisphere and then sum them up.
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Volume of the Hemisphere: The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]
According to the problem, the volume of the hemisphere is given as \( 4 , \text{in}^3 \).
Now we can set up the equation:
\[ \frac{2}{3} \pi r^3 = 4 \]
To find \( r^3 \), we rearrange this equation:
\[ r^3 = \frac{4 \cdot 3}{2 \pi} = \frac{12}{2 \pi} = \frac{6}{\pi} \]
Now we can find \( r \):
\[ r = \sqrt[3]{\frac{6}{\pi}} \]
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Volume of the Cone: The problem states that the diameter of the cone is equal to its height. The diameter \( d \) is equal to \( 2r \), where \( r \) is the radius we just found. Therefore, the height \( h \) of the cone is equal to \( d \), so \( h = 2r \).
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Plugging in \( h = 2r \):
\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Now substituting \( r^3 \):
\[ V = \frac{2}{3} \pi \left( \frac{6}{\pi} \right) = \frac{2 \cdot 6}{3} = 4 , \text{in}^3 \]
So the volume of the cone is also \( 4 , \text{in}^3 \).
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Total Volume: Now that we have the volumes of both the cone and the hemisphere, we can find the total volume of ice cream:
\[ \text{Total Volume} = \text{Volume of Cone} + \text{Volume of Hemisphere} = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]
Thus, the total volume of ice cream in and on top of the cone is \( \boxed{8} , \text{in}^3 \).