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 diameter of 5 cm. What is the volume of the hemisphere (half of a sphere), rounded to the nearest hundredth? *Show all steps to earn full credit.

Answer 1

To find the volume of the hemisphere, we can start by using the formula for the volume of a sphere and then divide it by 2 since a hemisphere is half of a sphere.

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

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

where \( r \) is the radius of the sphere.

1. **Determine the radius**: The problem states that the diameter of the hemisphere is 5 cm. The radius \( r \) is half of the diameter.

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

2. **Calculate the volume of the sphere using the radius**:

\[
V_{\text{sphere}} = \frac{4}{3} \pi (2.5)^3
\]

3. **Calculate \( (2.5)^3 \)**:

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

4. **Substitute \( (2.5)^3 \) back into the volume formula**:

\[
V_{\text{sphere}} = \frac{4}{3} \pi (15.625)
\]

5. **Now calculate \( \frac{4}{3} \times 15.625 \)**:

\[
\frac{4}{3} \times 15.625 = \frac{62.5}{3} \approx 20.8333
\]

6. **Now multiply by \( \pi \)**. Using \( \pi \approx 3.14159 \):

\[
V_{\text{sphere}} \approx 20.8333 \times 3.14159 \approx 65.497
\]

7. **Since we want the volume of the hemisphere, we divide the volume of the sphere by 2**:

\[
V_{\text{hemisphere}} = \frac{65.497}{2} \approx 32.7485
\]

8. **Round the answer to the nearest hundredth**:

\[
V_{\text{hemisphere}} \approx 32.75 \, \text{cm}^3
\]

Thus, the volume of the hemisphere, rounded to the nearest hundredth, is \( \boxed{32.75} \) cm³.