The surface area of sphere and total surface area of hemisphere is in ratio 3:1. What is the volume of sphere if sum of squares of radius of sphere and hemisphere is 13 cm (in `cm^(3)`)

Options

A. A)`72 pi`
B. B)`108 pi`
C. C)`32pi`
D. D)`36pi`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the given information
We know that the surface area of a sphere and the total surface area of a hemisphere are in the ratio of 3:1. We also know that the sum of the squares of the radius of the sphere (R) and the radius of the hemisphere (r) is 13 cm².

### Step 2: Write the formulas for surface areas
- The surface area of a sphere is given by the formula:
\[ \text{Surface Area of Sphere} = 4\pi R^2 \]
- The total surface area of a hemisphere is given by the formula:
\[ \text{Total Surface Area of Hemisphere} = 3\pi r^2 \]

### Step 3: Set up the ratio
According to the problem, we have:
\[ \frac{4\pi R^2}{3\pi r^2} = \frac{3}{1} \]
We can simplify this by canceling \(\pi\):
\[ \frac{4R^2}{3r^2} = 3 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 4R^2 = 9r^2 \]

### Step 5: Express \(R^2\) in terms of \(r^2\)
From the equation \(4R^2 = 9r^2\), we can express \(R^2\) as:
\[ R^2 = \frac{9}{4}r^2 \]

### Step 6: Use the sum of squares condition
We know that:
\[ R^2 + r^2 = 13 \]
Substituting \(R^2\) from the previous step:
\[ \frac{9}{4}r^2 + r^2 = 13 \]
Combine the terms:
\[ \frac{9}{4}r^2 + \frac{4}{4}r^2 = 13 \]
\[ \frac{13}{4}r^2 = 13 \]

### Step 7: Solve for \(r^2\)
Multiplying both sides by 4:
\[ 13r^2 = 52 \]
Dividing by 13:
\[ r^2 = 4 \]

### Step 8: Find \(R^2\)
Now substituting \(r^2\) back to find \(R^2\):
\[ R^2 = \frac{9}{4} \times 4 = 9 \]

### Step 9: Find \(R\)
Taking the square root of \(R^2\):
\[ R = 3 \text{ cm} \]

### Step 10: Calculate the volume of the sphere
The volume \(V\) of the sphere is given by:
\[ V = \frac{4}{3}\pi R^3 \]
Substituting \(R = 3\):
\[ V = \frac{4}{3}\pi (3)^3 = \frac{4}{3}\pi \times 27 = 36\pi \text{ cm}^3 \]

### Final Answer
The volume of the sphere is:
\[ \boxed{36\pi \text{ cm}^3} \]