A solid sphere of radius ‘a’ is cut by a plane at a distance of b’ from its centre, thus obtaining two different pieces. The total surface area of these two pieces is `140%` of the surface area of the sphere. Find ‘b’.

Options

A. `a/sqrt2`
B. `a/sqrt3`
C. `a/sqrt5`
D. `a/sqrt6`

Answer

To solve the problem, we need to find the distance \( b \) from the center of a solid sphere of radius \( a \) that is cut by a plane, resulting in two pieces whose total surface area is 140% of the original sphere's surface area.

### Step-by-Step Solution:

1. **Calculate the Surface Area of the Sphere**:
The surface area \( S \) of a sphere with radius \( a \) is given by the formula:
\[ S = 4\pi a^2 \]

2. **Determine the Total Surface Area of the Two Pieces**:
According to the problem, the total surface area of the two pieces after the cut is 140% of the original surface area:
\[ \text{Total Surface Area} = 1.4 \times S = 1.4 \times 4\pi a^2 = 5.6\pi a^2 \]

3. **Identify the Components of the Total Surface Area**:
When the sphere is cut by a plane, it creates two hemispherical pieces and two circular faces (the cut surfaces). The total surface area of the two pieces can be expressed as:
\[ \text{Total Surface Area} = \text{Curved Surface Area of both hemispheres} + \text{Area of the two circular faces} \]
The curved surface area of the two hemispheres is:
\[ 2 \times 2\pi a^2 = 4\pi a^2 \]
The area of the two circular faces (with radius \( r \)) is:
\[ 2 \times \pi r^2 \]
where \( r \) is the radius of the circular face formed by the cut.

4. **Relate the Radius \( r \) to \( a \) and \( b \)**:
Using the right triangle formed by the radius of the sphere, the distance \( b \) from the center to the plane, and the radius \( r \) of the circular face:
\[ r^2 + b^2 = a^2 \implies r = \sqrt{a^2 - b^2} \]

5. **Substitute \( r \) into the Total Surface Area Equation**:
Now, substituting \( r \) into the area of the circular faces:
\[ \text{Area of the two circular faces} = 2 \times \pi (a^2 - b^2) = 2\pi (a^2 - b^2) \]
Therefore, the total surface area becomes:
\[ 4\pi a^2 + 2\pi (a^2 - b^2) = 4\pi a^2 + 2\pi a^2 - 2\pi b^2 = 6\pi a^2 - 2\pi b^2 \]

6. **Set Up the Equation**:
We know that this total surface area equals \( 5.6\pi a^2 \):
\[ 6\pi a^2 - 2\pi b^2 = 5.6\pi a^2 \]

7. **Simplify the Equation**:
Dividing through by \( \pi \):
\[ 6a^2 - 2b^2 = 5.6a^2 \]
Rearranging gives:
\[ 6a^2 - 5.6a^2 = 2b^2 \implies 0.4a^2 = 2b^2 \implies b^2 = 0.2a^2 \]

8. **Solve for \( b \)**:
Taking the square root of both sides:
\[ b = a \sqrt{0.2} = \frac{a}{\sqrt{5}} \]

### Final Answer:
Thus, the value of \( b \) is:
\[ b = \frac{a}{\sqrt{5}} \]