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? use pi instead of 3.14



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Answer 1

To find the volume of a hemisphere, we can use the formula for the volume of a sphere and then take half of it since a hemisphere is half of a sphere.

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.

1. **Find the radius:**

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

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

2. **Calculate the volume of the whole sphere:**

Now, we can substitute the radius into the volume formula for the sphere:

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

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

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

Now plug this value back into the volume formula:

\[
V = \frac{4}{3} \pi (15.625) = \frac{62.5}{3} \pi
\]

3. **Calculate the volume of the hemisphere:**

To find the volume of the hemisphere, we take half of the volume of the sphere:

\[
V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{62.5}{3} \pi\right) = \frac{62.5}{6} \pi
\]

4. **Convert to a numerical value using \( \pi \):**

Using \( \pi \approx 3.14159265 \):

\[
V_{\text{hemisphere}} = \frac{62.5}{6} \times 3.14159265
\]

Now calculate \( \frac{62.5}{6} \):

\[
\frac{62.5}{6} = 10.41666666
\]

Now, multiply this by \( \pi \):

\[
V_{\text{hemisphere}} \approx 10.41666666 \times 3.14159265 \approx 32.769
\]

5. **Round to the nearest hundredth:**

Finally, rounding \( 32.769 \) to the nearest hundredth gives:

\[
V_{\text{hemisphere}} \approx 32.77 \, \text{cm}^3
\]

**Final answer: The volume of the hemisphere is approximately \( \boxed{32.77} \, \text{cm}^3 \).**