The ratio of the volume to surface area of the sphere is 8:3. What is the total surface area of the hemisphere, given that the radius of sphere and hemisphere is same? (in `unit^(2)`)

Options

A. `128pi`
B. `192pi`
C. `64pi`
D. `256pi`

Answer

To solve the problem step by step, we need to find the total surface area of a hemisphere given the ratio of the volume to surface area of a sphere.

### Step 1: Understand the given ratio
The problem states that the ratio of the volume to the surface area of the sphere is 8:3. This can be expressed mathematically as:
\[ \frac{\text{Volume of sphere}}{\text{Surface area of sphere}} = \frac{8}{3} \]

### Step 2: Write the formulas for volume and surface area of a sphere
The formulas for the volume \( V \) and surface area \( A \) of a sphere with radius \( r \) are:
- Volume of sphere:
\[ V = \frac{4}{3} \pi r^3 \]
- Surface area of sphere:
\[ A = 4 \pi r^2 \]

### Step 3: Set up the equation using the ratio
Using the formulas, we can set up the equation:
\[ \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = \frac{8}{3} \]

### Step 4: Simplify the equation
We can simplify the left side:
\[ \frac{\frac{4}{3} \pi r^3}{4 \pi r^2} = \frac{4}{3} \cdot \frac{r^3}{4 r^2} = \frac{r}{3} \]
So we have:
\[ \frac{r}{3} = \frac{8}{3} \]

### Step 5: Solve for \( r \)
To find \( r \), we can multiply both sides by 3:
\[ r = 8 \]

### Step 6: Find the total surface area of the hemisphere
The total surface area \( A_h \) of a hemisphere (including the base) is given by:
\[ A_h = 3 \pi r^2 \]
Now substituting \( r = 8 \):
\[ A_h = 3 \pi (8^2) = 3 \pi (64) = 192 \pi \]

### Final Answer
Thus, the total surface area of the hemisphere is:
\[ 192 \pi \text{ unit}^2 \]