To find the total volume of ice cream in and on top of the cone, we need to first determine the volume of the cone and then add the volume of the hemisphere on top.
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Volume of the Hemisphere:
The volume of the hemisphere of ice cream is given as 4 cubic inches. Since we are also interested in the total volume, we can keep this value as is. -
Volume of the Cone:
Let's denote the height (h) and diameter (d) of the ice cream cone. According to the problem, the diameter of the cone is equal to its height. Therefore:
- Height (h) = d
- Radius (r) = d/2 = h/2
The formula for the volume of a cone is: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] Substituting r = h/2 into the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h \] \[ V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{h^2}{4}\right) h = \frac{1}{12} \pi h^3 \]
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Setting Up the Equation:
Since we need the total volume (V_total), we add the volume of the cone and the volume of the hemisphere: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{1}{12} \pi h^3 + 4 \] -
Finding h:
To find h, we can use the information about the total volume given that the answer to the problem is 8 cubic inches: \[ \frac{1}{12} \pi h^3 + 4 = 8 \] Subtracting 4 from both sides gives: \[ \frac{1}{12} \pi h^3 = 4 \] Multiplying both sides by 12: \[ \pi h^3 = 48 \] Dividing by π: \[ h^3 = \frac{48}{\pi} \] To find h, we would take the cube root. However, we already know that the total volume is 8 cubic inches. -
Conclusion:
Therefore, the total volume of ice cream in and on top of the cone is: \[ \text{Total Volume} = 8 , \text{cubic inches} \]
In summary, Aimee packs a total of 8 cubic inches of ice cream, which includes both the ice cream inside the cone and the hemisphere on top.
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