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.
- Find the radius:
Given that the diameter of the hemisphere is 5 cm, we can find the radius \( r \) as follows:
\[ r = \frac{\text{diameter}}{2} = \frac{5 , \text{cm}}{2} = 2.5 , \text{cm} \]
- 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:
\[ V = \frac{4}{3} \pi (15.625) = \frac{62.5}{3} \pi \]
- Calculate the volume of the hemisphere:
To find the volume of the hemisphere, we take half of the volume of the sphere:
\[ V_{\text{hemisphere}} = \frac{1}{2} \left(\frac{62.5}{3} \pi\right) = \frac{62.5}{6} \pi \]
- 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 \( \pi \):
\[ V_{\text{hemisphere}} \approx 10.41666666 \times 3.14159265 \approx 32.769 \]
- 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 \( \boxed{32.77} , \text{cm}^3 \).
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