To solve the problem, we need to find the total volume of ice cream in and on top of the cone. The total volume will consist of the volume of the ice cream cone plus the volume of the hemisphere on top.
Step 1: Volume of the Hemisphere We know that the volume of the hemisphere of ice cream on top of the cone is given as:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
We are given that this volume is \(4 , \text{in}^3\):
\[ \frac{2}{3} \pi r^3 = 4 \]
To find \(r\), we can solve for \(r^3\):
\[ r^3 = \frac{4 \cdot 3}{2\pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Next, we calculate \(r\):
\[ r = \left(\frac{6}{\pi}\right)^{1/3} \]
Step 2: Volume of the Ice Cream Cone We also know that the ice cream cone has a height \(h\) equal to its diameter, which is \(2r\). Therefore, the height of the cone is:
\[ h = 2r \]
The volume of a cone is given by the formula:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting for \(h\):
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
From step 1, we found that:
\[ r^3 = \frac{6}{\pi} \] Thus:
\[ V_{\text{cone}} = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2}{3} \cdot \frac{12}{\pi} = \frac{24}{3\pi} = \frac{8}{\pi} \]
Step 3: Total Volume of Ice Cream Finally, we add the volumes of the hemisphere and the cone:
\[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{8}{\pi} + 4 \]
To add these volumes, we can rewrite \(4\) with a common denominator of \(\pi\):
\[ 4 = \frac{4\pi}{\pi} \]
Now we can add the two volumes:
\[ V_{\text{total}} = \frac{8}{\pi} + \frac{4\pi}{\pi} = \frac{8 + 4\pi}{\pi} \]
Thus, the total volume of ice cream in and on top of the cone is:
\[ \boxed{\frac{8 + 4\pi}{\pi}} \]
This represents the simplest form of the total volume expressed in terms of \(\pi\).
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