A hemi-spherical bowl has negligible thickness and the length of its circumference is 198 cm. Find the capacity of the bowl.

Answer

To find the capacity of the hemispherical bowl, we will follow these steps:

### Step 1: Understand the relationship between circumference and radius
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius.

### Step 2: Substitute the given circumference
We know the circumference of the bowl is 198 cm. Therefore, we can set up the equation:
\[ 2\pi r = 198 \]

### Step 3: Solve for the radius \( r \)
To find the radius \( r \), we can rearrange the equation:
\[ r = \frac{198}{2\pi} \]
Using \( \pi \approx 3.14 \):
\[ r = \frac{198}{2 \times 3.14} = \frac{198}{6.28} \approx 31.5 \text{ cm} \]

### Step 4: Use the radius to find the volume of the hemisphere
The volume \( V \) of a hemisphere is given by the formula:
\[ V = \frac{2}{3}\pi r^3 \]
Substituting the value of \( r \):
\[ V = \frac{2}{3}\pi (31.5)^3 \]

### Step 5: Calculate \( (31.5)^3 \)
Calculating \( (31.5)^3 \):
\[ (31.5)^3 = 31.5 \times 31.5 \times 31.5 = 31.5 \times 992.25 \approx 31276.875 \]

### Step 6: Substitute back to find the volume
Now substituting this back into the volume formula:
\[ V = \frac{2}{3} \times \pi \times 31276.875 \]
Using \( \pi \approx 3.14 \):
\[ V \approx \frac{2}{3} \times 3.14 \times 31276.875 \approx \frac{2 \times 3.14 \times 31276.875}{3} \]
Calculating this gives:
\[ V \approx \frac{196,172.5}{3} \approx 65,390.83 \text{ cm}^3 \]

### Final Answer
The capacity of the hemispherical bowl is approximately \( 65,390.83 \) cm³.

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