The surface area of a sphere is 423.5 cm2 less than total surface area of a hemisphere. Ifratio between radius of hemisphere and sphere is 3:2, then find the radius of hemisphere?

Options

A. 5.5cm
B. 5 cm
C. 4cm
D. 7 cm

Answer

To solve the problem, we need to find the radius of the hemisphere given the relationship between the surface areas of a sphere and a hemisphere, along with the ratio of their radii.

### Step-by-Step Solution:

1. **Understand the Surface Area Formulas**:
- The total 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:
\[ \text{Surface Area of Hemisphere} = 3\pi r^2 \]

2. **Set Up the Relationship**:
- According to the problem, the surface area of the sphere is 423.5 cm² less than the total surface area of the hemisphere:
\[ 3\pi r_{h}^2 - 4\pi r_{s}^2 = 423.5 \]
- Here, \( r_h \) is the radius of the hemisphere and \( r_s \) is the radius of the sphere.

3. **Use the Given Ratio**:
- The ratio of the radius of the hemisphere to the sphere is 3:2. We can express this as:
\[ r_h = 3x \quad \text{and} \quad r_s = 2x \]

4. **Substitute the Radii into the Area Equation**:
- Substitute \( r_h \) and \( r_s \) into the surface area equation:
\[ 3\pi (3x)^2 - 4\pi (2x)^2 = 423.5 \]
- Simplifying this gives:
\[ 3\pi (9x^2) - 4\pi (4x^2) = 423.5 \]
\[ 27\pi x^2 - 16\pi x^2 = 423.5 \]
\[ 11\pi x^2 = 423.5 \]

5. **Isolate \( x^2 \)**:
- Divide both sides by \( 11\pi \):
\[ x^2 = \frac{423.5}{11\pi} \]
- Using \( \pi \approx \frac{22}{7} \):
\[ x^2 = \frac{423.5 \times 7}{11 \times 22} \]

6. **Calculate \( x^2 \)**:
- Calculate the right-hand side:
\[ x^2 = \frac{2964.5}{242} \approx 12.25 \]

7. **Find \( x \)**:
- Taking the square root gives:
\[ x = \sqrt{12.25} = 3.5 \]

8. **Calculate the Radius of the Hemisphere**:
- Now, substitute \( x \) back to find the radius of the hemisphere:
\[ r_h = 3x = 3 \times 3.5 = 10.5 \text{ cm} \]

### Final Answer:
The radius of the hemisphere is **10.5 cm**.