The surface area of a sphere is `1386 cm^(2)`. Its volume is

Options

A. `1617 cm^(3)`
B. `3234 cm^(3)`
C. `4158 cm^(2)`
D. `9702 cm^(2)`

Answer

To find the volume of a sphere given its surface area, we can follow these steps:

### Step 1: Write down the formula for the surface area of a sphere.
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Substitute the given surface area into the formula.
We know that the surface area \( S \) is \( 1386 \, \text{cm}^2 \). Therefore, we can write:
\[ 1386 = 4 \pi r^2 \]

### Step 3: Solve for \( r^2 \).
To find \( r^2 \), we can rearrange the equation:
\[ r^2 = \frac{1386}{4 \pi} \]
Now, substituting \( \pi \) with \( \frac{22}{7} \):
\[ r^2 = \frac{1386}{4 \times \frac{22}{7}} = \frac{1386 \times 7}{4 \times 22} \]

### Step 4: Calculate \( r^2 \).
Calculating the values:
\[ r^2 = \frac{1386 \times 7}{88} \]
Calculating \( 1386 \times 7 = 9702 \):
\[ r^2 = \frac{9702}{88} = 110.25 \]

### Step 5: Find \( r \).
Now, take the square root of \( r^2 \) to find \( r \):
\[ r = \sqrt{110.25} = 10.5 \, \text{cm} \]

### Step 6: Write down the formula for the volume of a sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 7: Substitute \( r \) into the volume formula.
Now substitute \( r = 10.5 \, \text{cm} \) into the volume formula:
\[ V = \frac{4}{3} \pi (10.5)^3 \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times (10.5)^3 \]

### Step 8: Calculate \( (10.5)^3 \).
Calculating \( (10.5)^3 \):
\[ (10.5)^3 = 1157.625 \]

### Step 9: Calculate the volume.
Now substitute back into the volume equation:
\[ V = \frac{4}{3} \times \frac{22}{7} \times 1157.625 \]
Calculating this step by step:
1. Calculate \( \frac{4 \times 22}{3 \times 7} = \frac{88}{21} \).
2. Now multiply \( \frac{88}{21} \times 1157.625 \).

Calculating:
\[ V \approx 4158 \, \text{cm}^3 \]

### Final Answer:
The volume of the sphere is approximately \( 4158 \, \text{cm}^3 \).

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