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 a hemisphere, we can use the formula for the volume of a sphere and then take half of that volume.

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 Aimee's hemisphere has a diameter of 5 cm, we can find the radius by dividing the diameter by 2:

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

Now, we can calculate the volume of the full sphere using the radius:

\[ V = \frac{4}{3} \pi (2.5)^3 \]

Calculating \( (2.5)^3 \):

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

Now substituting this value into the volume formula:

\[ V = \frac{4}{3} \pi \times 15.625 \]

Calculating \( \frac{4}{3} \times 15.625 \):

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

Now we can multiply by \( \pi \):

\[ V \approx 20.8333 \times \pi \approx 20.8333 \times 3.14159 \approx 65.4498 \]

Since we are interested in the volume of the hemisphere, we take half of this volume:

\[ V_{\text{hemisphere}} = \frac{1}{2} \times 65.4498 \approx 32.7249 \]

Finally, rounding to the nearest hundredth, the volume of the hemisphere is:

\[ \boxed{32.72} \text{ cm}^3 \]