Question
Aimee packs ice cream into an I re cream co e she then puts a perfect hemisphere of ice cream on top of the co e that has a diameter of 5 cm what is the volume of the hemisphere half of the sphere
Answers
Answered by
GPT-4o mini
To find the volume of a hemisphere, we use the formula for the volume of a sphere and then take half of that value. The formula for the volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere. Since we need the volume of a hemisphere, we take half of this volume:
\[
V_{\text{hemisphere}} = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3
\]
Given that the diameter of the hemisphere is 5 cm, we can find the radius \( r \):
\[
r = \frac{\text{diameter}}{2} = \frac{5 \text{ cm}}{2} = 2.5 \text{ cm}
\]
Now we can substitute this value of \( r \) into the formula for the volume of the hemisphere:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (2.5)^3
\]
Calculating \( (2.5)^3 \):
\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625
\]
Now substituting back into the volume formula:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (15.625)
\]
Calculating:
\[
V_{\text{hemisphere}} = \frac{31.25}{3} \pi \approx 10.41667 \pi
\]
Now approximating \( \pi \) as 3.14 for practical calculations:
\[
V_{\text{hemisphere}} \approx 10.41667 \times 3.14 \approx 32.76 \text{ cm}^3
\]
Therefore, the volume of the hemisphere of ice cream on top of the cone is approximately:
\[
\boxed{32.76} \text{ cm}^3
\]
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere. Since we need the volume of a hemisphere, we take half of this volume:
\[
V_{\text{hemisphere}} = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3
\]
Given that the diameter of the hemisphere is 5 cm, we can find the radius \( r \):
\[
r = \frac{\text{diameter}}{2} = \frac{5 \text{ cm}}{2} = 2.5 \text{ cm}
\]
Now we can substitute this value of \( r \) into the formula for the volume of the hemisphere:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (2.5)^3
\]
Calculating \( (2.5)^3 \):
\[
(2.5)^3 = 2.5 \times 2.5 \times 2.5 = 15.625
\]
Now substituting back into the volume formula:
\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (15.625)
\]
Calculating:
\[
V_{\text{hemisphere}} = \frac{31.25}{3} \pi \approx 10.41667 \pi
\]
Now approximating \( \pi \) as 3.14 for practical calculations:
\[
V_{\text{hemisphere}} \approx 10.41667 \times 3.14 \approx 32.76 \text{ cm}^3
\]
Therefore, the volume of the hemisphere of ice cream on top of the cone is approximately:
\[
\boxed{32.76} \text{ cm}^3
\]