If the surface area of a sphere is `(144 pi)m^(2)` then its volume is

Options

A. `(288pi)m^(3)`
B. `(188 pi)m^(3)`
C. `(300 pi)m^(3)`
D. `(316pi) m^(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 the surface area is \( 144\pi \, m^2 \). Therefore, we can set up the equation:
\[ 4\pi r^2 = 144\pi \]

### Step 3: Simplify the equation.
We can divide both sides of the equation by \( \pi \) (since \( \pi \) is a common factor):
\[ 4r^2 = 144 \]

### Step 4: Solve for \( r^2 \).
Next, divide both sides by 4:
\[ r^2 = \frac{144}{4} = 36 \]

### Step 5: Find the radius \( r \).
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{36} = 6 \, m \]

### Step 6: Use the radius to find the volume.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]
Substituting \( r = 6 \, m \):
\[ V = \frac{4}{3}\pi (6)^3 \]

### Step 7: Calculate \( (6)^3 \).
Calculating \( 6^3 \):
\[ 6^3 = 216 \]

### Step 8: Substitute back into the volume formula.
Now substitute \( 216 \) back into the volume formula:
\[ V = \frac{4}{3}\pi (216) \]

### Step 9: Simplify the volume.
Now calculate:
\[ V = \frac{4 \times 216}{3}\pi = \frac{864}{3}\pi = 288\pi \, m^3 \]

### Final Answer:
Thus, the volume of the sphere is:
\[ V = 288\pi \, m^3 \]

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