To solve for the total volume of ice cream in and on top of the cone, we need to consider both the volume of the ice cream in the cone and the volume of the hemisphere on top of it.
Step 1: Calculate the volume of the hemisphere
Aimee puts a hemisphere of ice cream on top of the cone with a volume of \(4 , \text{in}^3\).
The formula for the volume of a full sphere is given by: \[ V = \frac{4}{3} \pi r^3 \] Since a hemisphere is half of a sphere, the volume \( V_h \) of a hemisphere is: \[ V_h = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \]
Setting this equal to the volume of the hemisphere given in the problem: \[ \frac{2}{3} \pi r^3 = 4 \] Now we can solve for \( r \): \[ r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Taking the cube root, we have: \[ r = \sqrt[3]{\frac{6}{\pi}} \]
Step 2: Calculate the volume of the cone
The diameter of the ice cream cone is equal to its height. Let \( h \) be the height of the cone. Since the diameter is equal to the radius times 2, the radius \( R \) of the cone can be defined as: \[ R = \frac{h}{2} \]
The formula for the volume of a cone is given by: \[ V = \frac{1}{3} \pi R^2 h \]
Substituting \( R = \frac{h}{2} \) into the volume equation: \[ V = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h = \frac{1}{3} \pi \frac{h^2}{4} h = \frac{1}{12} \pi h^3 \]
Step 3: Equate the volume of the ice cream cone to the volume of the hemisphere
Since the total volume of the ice cream (hemisphere on the cone) must match the volume of the cone, we will express \( V \) in terms of \( h \) equal to \(4 , \text{in}^3\): \[ \frac{1}{12} \pi h^3 + 4 = \frac{1}{12} \pi h^3 + \frac{2}{3} \pi r^3 = 4 \]
Total Volume of Ice Cream
Now, let's find the total volume of ice cream: \[ V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{1}{12} \pi h^3 + 4 \]
A specific relation or volume value cannot be determined since it depends on the specifics of \( h \). However, if you find \( h \), you can define the cone's volume.
Conclusion
To get the specific numeric total volume will need to establish and substitute \( h \) with the underlaid criteria. Thus, we sum the two volumes.
- Calculate the volume of the cone,
- Add \(4, \text{in}^3\) for the hemisphere on top ensuring it equals the overall condition.
The total volume of ice cream, therefore, depends highly on the linear height \(h\) which could be derived further fitting the condition at usage without needing an excess detail unless presented.
Final Note
For a correspondingly calculated specific height the approach renders simple, yielding \( \text{Total Volume} = 4 \text{ in}^3 \text{ plus the derived cone's volume} \).