Question

Aimee packs ice cream into an ice cream cone she then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 inches the diameter of the ice cream cone is equal to this height what is the total volume of ice cream in and on top of the cone use the relationship between the formulas for the volumes of cones and spheres to help solve this problem show your work and explain your reasoning

Answer 1

To solve this problem, we need to find the total volume of ice cream that includes both the ice cream inside the cone as well as the hemisphere on top of the cone.

### Given:
1. The volume of the hemisphere of ice cream on top of the cone is \(4 \text{ in}^3\).
2. The diameter of the ice cream cone is equal to its height.

### Step 1: Determine the radius of the hemisphere

The formula for the volume of a hemisphere is given by:
\[
V = \frac{2}{3} \pi r^3
\]
Given that the volume \(V\) of the hemisphere is \(4 \text{ in}^3\), we can set up the equation:
\[
\frac{2}{3} \pi r^3 = 4
\]
To solve for \(r^3\), multiply both sides by \(\frac{3}{2}\):
\[
\pi r^3 = 6
\]
\[
r^3 = \frac{6}{\pi}
\]
Now we can find \(r\):
\[
r = \left(\frac{6}{\pi}\right)^{1/3}
\]

### Step 2: Diameter and height of the cone

Since the diameter of the hemisphere is the same as the height of the cone, we have:
\[
d = 2r
\]
Thus, the height \(h\) of the cone is also \(2r\).

### Step 3: Calculate the volume of the cone

The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting \(h = 2r\) into the volume formula, we get:
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
Substituting the expression we found for \(r^3\):
\[
V = \frac{2}{3} \pi \left( \frac{6}{\pi} \right) = \frac{2 \times 6}{3} = 4 \text{ in}^3
\]

### Step 4: Total volume of ice cream

Now, we find the total volume of ice cream, which includes both the volume of the ice cream inside the cone and the volume of the hemisphere on top:
\[
\text{Total Volume} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 \text{ in}^3 + 4 \text{ in}^3 = 8 \text{ in}^3
\]

### Conclusion

The total volume of ice cream in and on top of the cone is:
\[
\boxed{8 \text{ in}^3}
\]