The surface area of a sphere is `616 cm^(2)`. The volume of the sphere would be :

Options

A. `1437(1)/(3) cm^(3)`
B. `2100 cm^(3)`
C. `2500 cm^(3)`
D. `1225(3)/(5) cm^(3)`

Answer

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

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

### Step 2: Set the surface area equal to the given value.
We know that the surface area is \( 616 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 4\pi r^2 = 616 \]

### Step 3: Solve for \( r^2 \).
First, we can divide both sides of the equation by \( 4\pi \):
\[ r^2 = \frac{616}{4\pi} \]
Using \( \pi \approx \frac{22}{7} \), we substitute this value in:
\[ r^2 = \frac{616}{4 \times \frac{22}{7}} = \frac{616 \times 7}{88} \]
Now, simplify \( \frac{616}{88} \):
\[ \frac{616}{88} = 7 \]
Thus,
\[ r^2 = 7 \times 7 = 49 \]

### Step 4: Find the radius \( r \).
Taking the square root of both sides gives us:
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 5: Use the radius to find the volume of the sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]
Substituting \( r = 7 \, \text{cm} \) into the volume formula:
\[ V = \frac{4}{3}\pi (7)^3 \]
Calculating \( 7^3 \):
\[ 7^3 = 343 \]
Thus,
\[ V = \frac{4}{3}\pi \times 343 \]

### Step 6: Substitute \( \pi \) and calculate the volume.
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{4}{3} \times \frac{22}{7} \times 343 \]
The \( 7 \) in the denominator cancels with \( 343 \):
\[ 343 \div 7 = 49 \]
So we have:
\[ V = \frac{4}{3} \times 22 \times 49 \]
Calculating \( 22 \times 49 \):
\[ 22 \times 49 = 1078 \]
Now, substituting back:
\[ V = \frac{4 \times 1078}{3} = \frac{4312}{3} \approx 1437.33 \, \text{cm}^3 \]

### Final Answer:
The volume of the sphere is approximately \( 1437.33 \, \text{cm}^3 \).

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