Find the volume (in `cm^(3)`) of an hemisphere of diameter 21 cm.

Options

A. 2235.5
B. 2425.5
C. 2040
D. 1860

Answer

To find the volume of a hemisphere with a diameter of 21 cm, we will follow these steps:

### Step 1: Find the radius of the hemisphere
The radius \( r \) is half of the diameter. Given that the diameter is 21 cm:
\[ r = \frac{diameter}{2} = \frac{21 \, \text{cm}}{2} = 10.5 \, \text{cm} \]

### Step 2: Use the formula for the volume of a hemisphere
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3} \pi r^3 \]

### Step 3: Substitute the radius into the volume formula
Now we substitute \( r = 10.5 \, \text{cm} \) into the volume formula:
\[ V = \frac{2}{3} \pi (10.5)^3 \]

### Step 4: Calculate \( (10.5)^3 \)
First, calculate \( 10.5^3 \):
\[ 10.5^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 \]

### Step 5: Substitute \( (10.5)^3 \) back into the volume formula
Now we have:
\[ V = \frac{2}{3} \pi (1157.625) \]

### Step 6: Use \( \pi \approx \frac{22}{7} \) for calculation
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ V = \frac{2}{3} \times \frac{22}{7} \times 1157.625 \]

### Step 7: Simplify the expression
Calculating \( \frac{2 \times 22 \times 1157.625}{3 \times 7} \):
\[ = \frac{2 \times 22 \times 1157.625}{21} = \frac{51057.75}{21} \approx 2431.25 \, \text{cm}^3 \]

### Final Volume
Thus, the volume of the hemisphere is approximately:
\[ V \approx 2431.25 \, \text{cm}^3 \]