The total surface area of a sphere is `8pi` square unit. The volume of the sphere is

Options

A. `(8sqrt(2))/(3)pi` cubic unit
B. `(8)/(3) pi` cubic unit
C. `8sqrt(3)pi` cubic unit
D. `(8sqrt(3))/(5)pi` cubic unit

Answer

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

### Step 1: Understand the formula for the total surface area of a sphere.
The total surface area (TSA) of a sphere is given by the formula:
\[ \text{TSA} = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Set up the equation using the given total surface area.
We are given that the total surface area of the sphere is \( 8\pi \) square units. Therefore, we can set up the equation:
\[ 4\pi r^2 = 8\pi \]

### Step 3: Simplify the equation.
To simplify, we can divide both sides of the equation by \( 4\pi \):
\[ r^2 = \frac{8\pi}{4\pi} = 2 \]

### Step 4: Solve for the radius \( r \).
Now, we take the square root of both sides to find \( r \):
\[ r = \sqrt{2} \]

### Step 5: Use the radius to find the volume of the sphere.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]
Substituting \( r = \sqrt{2} \) into the volume formula:
\[ V = \frac{4}{3}\pi (\sqrt{2})^3 \]

### Step 6: Calculate \( (\sqrt{2})^3 \).
Calculating \( (\sqrt{2})^3 \):
\[ (\sqrt{2})^3 = \sqrt{2} \times \sqrt{2} \times \sqrt{2} = 2\sqrt{2} \]

### Step 7: Substitute back into the volume formula.
Now substitute \( 2\sqrt{2} \) back into the volume formula:
\[ V = \frac{4}{3}\pi (2\sqrt{2}) = \frac{8\sqrt{2}}{3}\pi \]

### Final Answer:
The volume of the sphere is:
\[ V = \frac{8\sqrt{2}}{3}\pi \text{ cubic units} \]

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