To find the volume of a hemisphere, we start by using the formula for the volume of a full sphere and then divide it by 2.
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.
Step 1: Find the radius of the hemisphere
Given that the diameter of the hemisphere is 5 cm, we can find the radius \( r \):
\[ r = \frac{\text{diameter}}{2} = \frac{5 \text{ cm}}{2} = 2.5 \text{ cm} \]
Step 2: Calculate the volume of the hemisphere
Since a hemisphere is half of a sphere, the formula for the volume of a hemisphere \( V_h \) is:
\[ V_h = \frac{1}{2} \times V_{\text{sphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 \]
Now substituting \( r \):
\[ V_h = \frac{1}{2} \times \frac{4}{3} \pi (2.5)^3 \]
Step 3: Calculate \( (2.5)^3 \)
Calculating \( (2.5)^3 \):
\[ (2.5)^3 = 2.5 \times 2.5 \times 2.5 \]
Calculating step by step:
- \( 2.5 \times 2.5 = 6.25 \)
- \( 6.25 \times 2.5 = 15.625 \)
So,
\[ (2.5)^3 = 15.625 \]
Step 4: Substitute back into the volume formula
Now, substitute back into the volume formula:
\[ V_h = \frac{1}{2} \times \frac{4}{3} \pi (15.625) \]
\[ = \frac{4 \times 15.625}{6} \pi \]
Calculating \( 4 \times 15.625 \):
\[ 4 \times 15.625 = 62.5 \]
So, we now have:
\[ V_h = \frac{62.5}{6} \pi \]
Step 5: Calculate \( \frac{62.5}{6} \)
Calculating \( \frac{62.5}{6} \):
\[ 62.5 \div 6 \approx 10.41667 \]
Step 6: Final volume calculation
Now substituting back to find the volume:
\[ V_h \approx 10.41667 \pi \]
Using \( \pi \approx 3.14159 \):
\[ V_h \approx 10.41667 \times 3.14159 \approx 32.749 \]
Step 7: Round to the nearest hundredth
Rounding \( 32.749 \) to the nearest hundredth:
\[ V_h \approx 32.75 \text{ cm}^3 \]
Final Answer
Therefore, the volume of the hemisphere rounded to the nearest hundredth is:
\[ \boxed{32.75} \text{ cm}^3 \]