To find the total volume of ice cream (both in the cone and on top of the cone), we need to calculate the volume of the cone and add the volume of the hemisphere.
Step 1: Understanding the Given Information
- The volume of the hemisphere is given as \(4 , \text{in}^3\).
- The cone's diameter is equal to its height. Denote the radius of the cone as \(r\) and the height as \(h\). Hence, \(h = 2r\) (since diameter = 2 × radius).
Step 2: Volume of the Hemisphere
The formula for the volume \(V\) of a hemisphere is: \[ V = \frac{2}{3} \pi r^3 \] Setting this equal to the given volume of the hemisphere: \[ \frac{2}{3} \pi r^3 = 4 \] To solve for \(r^3\): \[ r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi} \]
Step 3: Calculating the Volume of the Cone
The formula for the volume \(V\) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] Substituting \(h = 2r\) into the formula gives: \[ V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3 \] Now substituting \(r^3\) from earlier: \[ V = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = 4 \]
Step 4: Total Volume Calculation
Now, we can find the total volume of ice cream: \[ \text{Total Volume} = \text{Volume of Cone} + \text{Volume of Hemisphere} = 4 , \text{in}^3 + 4 , \text{in}^3 = 8 , \text{in}^3 \]
Conclusion
The total volume of ice cream in and on top of the cone is \(\boxed{8} , \text{in}^3\).