To find the total volume of ice cream in and on top of the cone, we need to calculate two volumes: the volume of the cone and the volume of the hemisphere on top.
First, let's analyze the volume of the hemisphere. The formula for the volume \(V\) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Given that the volume of the hemisphere is \(4 , \text{in}^3\), we can set up the equation:
\[ \frac{2}{3} \pi r^3 = 4 \]
To find \(r\), we solve for \(r^3\):
\[ r^3 = \frac{4 \cdot 3}{2\pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Now, we calculate \(r\):
\[ r = \left(\frac{6}{\pi}\right)^{\frac{1}{3}} \]
Next, we need to find the volume of the cone. The volume \(V_c\) of a cone is given by:
\[ V_c = \frac{1}{3} \pi r^2 h \]
We are informed that the diameter of the cone is equal to its height. Since the diameter is \(2r\), the height \(h\) can be expressed as:
\[ h = 2r \]
Now substituting into the volume formula for the cone:
\[ V_c = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
We already found \(r^3\) as \(\frac{6}{\pi}\), so now we can calculate \(V_c\):
\[ V_c = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = 4 , \text{in}^3 \]
Now we can find the total volume of ice cream, which is the sum of the volumes of the cone and the hemisphere:
\[ \text{Total Volume} = V_c + V_h = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]
Thus, the total volume of ice cream in and on top of the cone is
\[ \boxed{8 , \text{in}^3} \]
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