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 in.3
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

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

### Step 1: Find the radius of the hemisphere
We know that the volume of the hemisphere of ice cream is given as \(4 \, \text{in}^3\). The formula for the volume of a hemisphere is:

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

Setting this equal to the volume of the hemisphere we have:

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

Now, solve for \(r^3\):

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

\[
r^3 = \frac{6}{\pi}
\]

Now, calculate \(r\):

\[
r = \left(\frac{6}{\pi}\right)^{1/3}
\]

### Step 2: Calculate the volume of the cone
The diameter of the cone is equal to its height, which implies:

\[
d = h
\]

If we let \(r_c\) be the radius of the cone, the diameter \(d = 2r_c\) implies:

\[
h = 2r_c
\]

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

\[
V_c = \frac{1}{3} \pi r_c^2 h
\]

Substituting \(h = 2r_c\):

\[
V_c = \frac{1}{3} \pi r_c^2 (2r_c) = \frac{2}{3} \pi r_c^3
\]

### Step 3: Relating the cone's radius to the hemisphere's radius
Since the diameter of the cone is equal to its height, the radius of the cone can also be taken as the radius of the hemisphere that is resting on it, that is \(r_c = r\).

Substituting \(r_c = r\):

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

### Step 4: Combining the volumes
Now, substituting the previously calculated \(r^3\) into the cone volume formula:

\[
V_c = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2}{3} \cdot 6 = 4 \, \text{in}^3
\]

### Step 5: Total volume
Now to find the total volume of ice cream, we sum the volumes of the cone and the hemisphere:

\[
\text{Total Volume} = V_c + V_h = 4 \, \text{in}^3 + 4 \, \text{in}^3 = 8 \, \text{in}^3
\]

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

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