To calculate the total volume of ice cream in and on top of the cone, we'll need to find the volumes of both the cone and the hemisphere.
Step 1: Find the dimensions of the cone
Let's denote:
- The height of the cone \( h \).
- The diameter of the cone \( d \) (which is also equal to its height as per the problem).
Since the diameter is equal to the height, we have: \[ d = h \] The radius \( r \) of the cone is then: \[ r = \frac{d}{2} = \frac{h}{2} \]
Step 2: Using the volume of the cone
The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \]
Substituting \( r = \frac{h}{2} \): \[ V = \frac{1}{3} \pi \left(\frac{h}{2}\right)^2 h = \frac{1}{3} \pi \frac{h^2}{4} h \] \[ V = \frac{1}{3} \cdot \frac{\pi}{4} h^3 = \frac{\pi}{12} h^3 \]
Step 3: Set the volume of the cone to \( 4 , \text{in}^3 \)
According to the problem, the volume of ice cream in the cone is \( 4 , \text{in}^3 \): \[ \frac{\pi}{12} h^3 = 4 \]
Step 4: Solve for \( h \)
Now, let's isolate \( h^3 \): \[ h^3 = 4 \cdot \frac{12}{\pi} = \frac{48}{\pi} \]
Now take the cube root: \[ h = \left(\frac{48}{\pi}\right)^{1/3} \]
Step 5: Find the radius of the hemisphere
The radius of the hemisphere is equal to the radius of the cone, which has been derived as: \[ r = \frac{h}{2} = \frac{1}{2} \left(\frac{48}{\pi}\right)^{1/3} \]
Step 6: Volume of the hemisphere
The volume \( V_h \) of a hemisphere is given by: \[ V_h = \frac{2}{3} \pi r^3 \]
Substituting the radius: \[ V_h = \frac{2}{3} \pi \left(\frac{h}{2}\right)^3 = \frac{2}{3} \pi \left(\frac{1}{2} \left(\frac{48}{\pi}\right)^{1/3}\right)^3 \] \[ = \frac{2}{3} \pi \left(\frac{1}{8} \cdot \left(\frac{48}{\pi}\right)} = \frac{2}{3} \cdot \frac{\pi}{8} \cdot \frac{48}{\pi} \] \[ = \frac{2}{3} \cdot 6 = 4 , \text{in}^3 \]
Step 7: Total Volume of Ice Cream
Now, adding the volume inside and on top of the cone: \[ \text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]
Conclusion
Therefore, the total volume of ice cream in and on top of the cone is: \[ \boxed{8 , \text{in}^3} \]