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?

Answer 1

To find the volume of a hemisphere, we first need to determine the formula for the volume of a sphere and then modify it for a hemisphere.

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 are dealing with a hemisphere, the volume of the hemisphere \( V_h \) is:

\[
V_h = \frac{1}{2} V = \frac{1}{2} \left( \frac{4}{3} \pi r^3 \right) = \frac{2}{3} \pi r^3
\]

Next, we need to find the radius of the hemisphere. Given that the diameter of the hemisphere is 5 cm, we can find the radius:

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

Now, we can substitute the radius back into the formula for the volume of the hemisphere:

\[
V_h = \frac{2}{3} \pi (2.5)^3
\]

Calculating \( (2.5)^3 \):

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

Now we substitute this value back into the volume formula:

\[
V_h = \frac{2}{3} \pi \times 15.625
\]

Now we calculate \( \frac{2}{3} \times 15.625 \):

\[
\frac{2}{3} \times 15.625 = \frac{31.25}{3} \approx 10.4167
\]

Now multiply by \( \pi \):

\[
V_h \approx 10.4167 \pi
\]

Using \( \pi \approx 3.14159 \):

\[
V_h \approx 10.4167 \times 3.14159 \approx 32.707 \text{ cm}^3
\]

Finally, rounding to the nearest hundredth:

\[
V_h \approx 32.71 \text{ cm}^3
\]

Thus, the volume of the hemisphere is approximately \( \boxed{32.71} \) cm³.