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 diameter of 5 cm.

What is the volume of the hemisphere (half of a sphere), rounded to the nearest hundredth?

1 answer

To find the volume of the hemisphere of ice cream, we first need to understand the formula for the volume of a sphere.

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

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

Since we need the volume of a hemisphere, we take half of the sphere's volume:

\[ V_{\text{hemisphere}} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \]

Given that the diameter of the hemisphere is 5 cm, we find the radius \( r \):

\[ r = \frac{\text{diameter}}{2} = \frac{5}{2} = 2.5 \text{ cm} \]

Now we can calculate the volume of the hemisphere:

\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (2.5)^3 \]

First, calculate \( (2.5)^3 \):

\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 = 6.25 \times 2.5 = 15.625 \]

Now substitute this back into the volume formula:

\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (15.625) \]

Now compute:

\[ \frac{2}{3} \times 15.625 = \frac{31.25}{3} \approx 10.41666667 \]

Now multiply by \( \pi \):

\[ V_{\text{hemisphere}} \approx 10.41666667 \times \pi \approx 10.41666667 \times 3.14159265 \approx 32.707 \quad \text{(approximately)} \]

Finally, round to the nearest hundredth:

\[ \boxed{32.71} \text{ cm}^3 \]