Question
There is an ice cream cone that has a perfect hemisphere on top of a cone. The hemisphere has a volume of 4in^3. The diameter of the ice cream cone is equal to its height. What is the total volume of the ice cream in and on top of the cone? show your work
Answers
GPT-4o mini
To find the total volume of the ice cream in and on top of the cone, we first need to determine the volume of the ice cream in the hemisphere and then find the volume of the cone below it.
### Step 1: Volume of the Hemisphere
The formula for the volume of a hemisphere is given by:
\[
V = \frac{2}{3} \pi r^3
\]
We are given that the volume of the hemisphere is \(4 \, \text{in}^3\). Setting the volume formula equal to \(4\):
\[
\frac{2}{3} \pi r^3 = 4
\]
To solve for \(r^3\), we first multiply both sides by \(3\):
\[
2\pi r^3 = 12
\]
Next, divide both sides by \(2\pi\):
\[
r^3 = \frac{12}{2\pi} = \frac{6}{\pi}
\]
Now we take the cube root of both sides to find \(r\):
\[
r = \sqrt[3]{\frac{6}{\pi}}
\]
### Step 2: Height and Diameter of the Cone
We know the diameter of the hemisphere is \(2r\). Therefore, the diameter of the cone is also \(2r\), and since the height of the cone is equal to its diameter, we have:
\[
\text{Height of the cone} = 2r
\]
### Step 3: Volume of the Cone
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting \(h = 2r\):
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
### Step 4: Substitute for \(r^3\)
From Step 1, we found that \(r^3 = \frac{6}{\pi}\). So now we substitute this value back into the volume formula for the cone:
\[
V = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = \frac{12}{3} = 4 \, \text{in}^3
\]
### Step 5: Total Volume of Ice Cream
Finally, to find the total volume of the ice cream both in the hemisphere and in the cone, we simply add the two volumes together:
\[
\text{Total Volume} = \text{Volume of Hemisphere} + \text{Volume of Cone}
\]
\[
\text{Total Volume} = 4 + 4 = 8 \, \text{in}^3
\]
Thus, the total volume of the ice cream in and on top of the cone is:
\[
\boxed{8 \, \text{in}^3}
\]
### Step 1: Volume of the Hemisphere
The formula for the volume of a hemisphere is given by:
\[
V = \frac{2}{3} \pi r^3
\]
We are given that the volume of the hemisphere is \(4 \, \text{in}^3\). Setting the volume formula equal to \(4\):
\[
\frac{2}{3} \pi r^3 = 4
\]
To solve for \(r^3\), we first multiply both sides by \(3\):
\[
2\pi r^3 = 12
\]
Next, divide both sides by \(2\pi\):
\[
r^3 = \frac{12}{2\pi} = \frac{6}{\pi}
\]
Now we take the cube root of both sides to find \(r\):
\[
r = \sqrt[3]{\frac{6}{\pi}}
\]
### Step 2: Height and Diameter of the Cone
We know the diameter of the hemisphere is \(2r\). Therefore, the diameter of the cone is also \(2r\), and since the height of the cone is equal to its diameter, we have:
\[
\text{Height of the cone} = 2r
\]
### Step 3: Volume of the Cone
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting \(h = 2r\):
\[
V = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3
\]
### Step 4: Substitute for \(r^3\)
From Step 1, we found that \(r^3 = \frac{6}{\pi}\). So now we substitute this value back into the volume formula for the cone:
\[
V = \frac{2}{3} \pi \left(\frac{6}{\pi}\right) = \frac{2 \cdot 6}{3} = \frac{12}{3} = 4 \, \text{in}^3
\]
### Step 5: Total Volume of Ice Cream
Finally, to find the total volume of the ice cream both in the hemisphere and in the cone, we simply add the two volumes together:
\[
\text{Total Volume} = \text{Volume of Hemisphere} + \text{Volume of Cone}
\]
\[
\text{Total Volume} = 4 + 4 = 8 \, \text{in}^3
\]
Thus, the total volume of the ice cream in and on top of the cone is:
\[
\boxed{8 \, \text{in}^3}
\]