To solve for the total volume of ice cream in and on top of the cone, we need to calculate both the volume of the cone of ice cream and the volume of the hemisphere on top.
Step 1: Volume of the Hemisphere
We know that the volume of the hemisphere is given as 4 in³.
Step 2: Volume of the Cone
We also know that the diameter of the ice cream cone is equal to its height. Let's denote:
- \( r \): the radius of the cone
- \( h \): the height of the cone
Since the diameter is equal to the height, we can write: \[ d = 2r = h \] Thus, \[ h = 2r \]
The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \( h = 2r \): \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \]
Step 3: Finding the Radius
Next, we need to relate the radius of the hemisphere to the radius of the cone. The volume of a hemisphere is given by: \[ V_{\text{hemisphere}} = \frac{2}{3} \pi r_{\text{hemisphere}}^3 \] Setting the volume of the hemisphere equal to 4 in³: \[ \frac{2}{3} \pi r_{\text{hemisphere}}^3 = 4 \] Solving for \( r_{\text{hemisphere}}^3 \): \[ r_{\text{hemisphere}}^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \] Now we get: \[ r_{\text{hemisphere}} = \left( \frac{6}{\pi} \right)^{1/3} \]
Step 4: Relating Hemisphere Radius to Cone Dimensions
Since the top hemisphere rests on the cone, its radius must equal the radius of the cone. Therefore: \[ r = r_{\text{hemisphere}} = \left( \frac{6}{\pi} \right)^{1/3} \]
Step 5: Volume of the Cone
Now, we can find the volume of the cone: \[ V_{\text{cone}} = \frac{2}{3} \pi r_{\text{hemisphere}}^3 \] Since we found that \( r_{\text{hemisphere}}^3 = \frac{6}{\pi} \): \[ V_{\text{cone}} = \frac{2}{3} \pi \left(\left(\frac{6}{\pi}\right)^{1/3}\right)^3 = \frac{2}{3} \pi \cdot \frac{6}{\pi} = \frac{2 \cdot 6}{3} = 4 \] Thus, the volume of the cone is 4 in³.
Step 6: Total Volume of Ice Cream
Finally, the total volume of ice cream includes both the cone and the hemisphere: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 + 4 = 8 \text{ in}^3 \]
Conclusion
The total volume of ice cream in and on top of the cone is 8 cubic inches.
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