Question

imee 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 in.3 The diameter of the ice cream cone is equal to its 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

Make it shorter and much mroe simple please no long smart explanations

Answer 2

To find the total volume of ice cream in and on top of the cone, we need to calculate the volume of the cone and add it to the volume of the hemisphere.

### Step 1: Calculate the Volume of the Hemisphere

We're given that the volume of the hemisphere is 4 cubic inches. The volume \( V \) of a hemisphere is related to its radius \( r \) by the formula:

\[
V = \frac{2}{3} \pi r^3
\]

Given that the volume of the hemisphere is 4 in³:

\[
\frac{2}{3} \pi r^3 = 4
\]

To find \( r^3 \), we rearrange the equation:

\[
r^3 = \frac{4 \times 3}{2 \pi} = \frac{12}{2\pi} = \frac{6}{\pi}
\]

### Step 2: Calculate the Diameter of the Hemisphere

Since the diameter \( D \) is twice the radius, we have:

\[
D = 2r
\]

### Step 3: Relate Cone Dimensions to Hemisphere

According to the problem, the diameter of the cone is equal to its height. Let's denote \( h \) as the height of the cone. The diameter of the cone \( D \) is:

\[
D = 2r
\]

Since the diameter of the cone equals its height \( h \):

\[
h = D = 2r
\]

### Step 4: Express the Height in Terms of r

So we need \( h \) in terms of \( r \):

Given \( D = 2r \),

\[
h = 2r
\]

### Step 5: Calculate the Volume of the Cone

The volume \( V \) of a cone is given by the formula:

\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]

We also know that \( h = 2r \). Substituting this into the cone volume formula provides:

\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]

### Step 6: Use the Volume of the Hemisphere to Find Cone Volume

From our earlier work, we know:

\[
\frac{2}{3} \pi r^3 = 4 \quad \Rightarrow \quad V_{\text{cone}} = 4
\]

### Step 7: Find Total Volume

Now, we can find the total volume of the ice cream, which is the sum of the volume of the ice cream cone and the volume of the hemisphere:

\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 + 4 = 8 \text{ in}^3
\]

### Conclusion

The total volume of ice cream in and on top of the cone is:

\[
\boxed{8 \text{ in}^3}
\]

Answer 3

Sure! Here’s a simplified version:

1. **Hemisphere Volume**: \( V = 4 \, \text{in}^3 \)

2. **Cone Volume**: The volume of the cone \( V_{\text{cone}} \) is the same as the volume of the hemisphere using the relationship:
\[
V_{\text{cone}} = 4 \, \text{in}^3
\]

3. **Total Volume**:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}} = 4 + 4 = 8 \, \text{in}^3
\]

Thus, the total volume of ice cream is:
\[
\boxed{8 \, \text{in}^3}
\]