To find the total volume of ice cream in and on top of the cone, we'll work through the problems step by step.
Step 1: Volume of the Hemisphere
The problem states that the volume of the hemisphere of ice cream on top of the cone is 4 in³. The formula for the volume of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Where \( r \) is the radius of the hemisphere. Since we know the volume is 4 in³, we can solve for \( r \):
\[ 4 = \frac{2}{3} \pi r^3 \]
To solve for \( r^3 \), we first multiply both sides by \( \frac{3}{2} \):
\[ r^3 = \frac{4 \cdot 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Now we can find the radius \( r \):
\[ r = \left(\frac{6}{\pi}\right)^{1/3} \]
Step 2: Dimensions of the Cone
The problem states that the diameter of the cone is equal to its height. The diameter \( d \) is twice the radius \( r \), and the formula for the radius of the hemisphere on the cone gives us:
\[ d = 2r \]
Since the diameter is equal to the height \( h \) of the cone:
\[ h = d = 2r \]
So, the radius of the cone \( R \) is:
\[ R = \frac{d}{2} = r \]
Thus, the height \( h \) of the cone can be expressed as:
\[ h = 2r \]
Step 3: Volume of the Ice Cream Cone
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi R^2 h \]
Substituting \( R \) and \( h \):
\[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Using our previous calculation for \( r^3 \):
\[ V = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = 4 \text{ in}^3 \]
Step 4: Total Volume of Ice Cream
Now, we need to calculate the total volume of ice cream, which is the sum of the volume in the cone and the volume on top:
The volume in the cone is also 4 in³. Thus, the total volume of ice cream is:
\[ \text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 + 4 = 8 \text{ in}^3 \]
Final Answer:
The total volume of ice cream in and on top of the cone is:
\[ \boxed{8 \text{ in}^3} \]
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