Question
If the ice cream cone below is totally full of ice cream, and the scoop on top is a perfectly round hemisphere, what is the total volume of ice cream, rounded to the nearest cm? Show all your work to get full credit for your answer.
the cone has a radius of 4 cm and a hight of 8 cm
Answers
GPT-4o mini
To find the total volume of ice cream in the cone and the hemisphere on top, we will calculate the volume of the cone and the volume of the hemisphere separately, then add them together.
### Step 1: Volume of the Cone
The volume \( V \) of a cone can be calculated using the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.
Given:
- Radius \( r = 4 \) cm
- Height \( h = 8 \) cm
Substituting the values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi (4^2)(8)
\]
Calculating \( 4^2 \):
\[
4^2 = 16
\]
Now substitute back into the equation:
\[
V_{\text{cone}} = \frac{1}{3} \pi (16)(8)
\]
Calculating \( 16 \times 8 \):
\[
16 \times 8 = 128
\]
Now substitute back into the equation:
\[
V_{\text{cone}} = \frac{1}{3} \pi (128)
\]
Calculating \( \frac{128}{3} \):
\[
\frac{128}{3} \approx 42.67
\]
Thus:
\[
V_{\text{cone}} \approx 42.67 \pi
\]
Using \( \pi \approx 3.14 \):
\[
V_{\text{cone}} \approx 42.67 \times 3.14 \approx 134.5 \, \text{cm}^3
\]
### Step 2: Volume of the Hemisphere
The volume \( V \) of a hemisphere can be calculated with the formula:
\[
V = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere.
Given:
- Radius \( r = 4 \) cm
Substituting the values into the formula:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (4^3)
\]
Calculating \( 4^3 \):
\[
4^3 = 64
\]
Now substitute back into the equation:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (64)
\]
Calculating \( \frac{2 \times 64}{3} \):
\[
\frac{128}{3} \approx 42.67
\]
Thus:
\[
V_{\text{hemisphere}} \approx 42.67 \pi
\]
Using \( \pi \approx 3.14 \):
\[
V_{\text{hemisphere}} \approx 42.67 \times 3.14 \approx 134.5 \, \text{cm}^3
\]
### Step 3: Total Volume of Ice Cream
Now, we add the volumes of the cone and the hemisphere:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}}
\]
Substituting the approximate values we calculated:
\[
V_{\text{total}} \approx 134.5 + 134.5 \approx 269 \, \text{cm}^3
\]
### Final Answer
Thus, the total volume of ice cream in the cone and the scoop is approximately:
\[
\boxed{269 \, \text{cm}^3}
\]
### Step 1: Volume of the Cone
The volume \( V \) of a cone can be calculated using the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.
Given:
- Radius \( r = 4 \) cm
- Height \( h = 8 \) cm
Substituting the values into the formula:
\[
V_{\text{cone}} = \frac{1}{3} \pi (4^2)(8)
\]
Calculating \( 4^2 \):
\[
4^2 = 16
\]
Now substitute back into the equation:
\[
V_{\text{cone}} = \frac{1}{3} \pi (16)(8)
\]
Calculating \( 16 \times 8 \):
\[
16 \times 8 = 128
\]
Now substitute back into the equation:
\[
V_{\text{cone}} = \frac{1}{3} \pi (128)
\]
Calculating \( \frac{128}{3} \):
\[
\frac{128}{3} \approx 42.67
\]
Thus:
\[
V_{\text{cone}} \approx 42.67 \pi
\]
Using \( \pi \approx 3.14 \):
\[
V_{\text{cone}} \approx 42.67 \times 3.14 \approx 134.5 \, \text{cm}^3
\]
### Step 2: Volume of the Hemisphere
The volume \( V \) of a hemisphere can be calculated with the formula:
\[
V = \frac{2}{3} \pi r^3
\]
where \( r \) is the radius of the hemisphere.
Given:
- Radius \( r = 4 \) cm
Substituting the values into the formula:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (4^3)
\]
Calculating \( 4^3 \):
\[
4^3 = 64
\]
Now substitute back into the equation:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (64)
\]
Calculating \( \frac{2 \times 64}{3} \):
\[
\frac{128}{3} \approx 42.67
\]
Thus:
\[
V_{\text{hemisphere}} \approx 42.67 \pi
\]
Using \( \pi \approx 3.14 \):
\[
V_{\text{hemisphere}} \approx 42.67 \times 3.14 \approx 134.5 \, \text{cm}^3
\]
### Step 3: Total Volume of Ice Cream
Now, we add the volumes of the cone and the hemisphere:
\[
V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}}
\]
Substituting the approximate values we calculated:
\[
V_{\text{total}} \approx 134.5 + 134.5 \approx 269 \, \text{cm}^3
\]
### Final Answer
Thus, the total volume of ice cream in the cone and the scoop is approximately:
\[
\boxed{269 \, \text{cm}^3}
\]