Study the following information carefully and answer the given questions: An apple pie of R cm radius has to cut into x identical pieces. Area of top surface of each piece was ` 0. 77 cm^(2)` but later on it was found `50%` of pie was rotten so the remaining `50%` was cut into (x - 3) pieces of area `0.616 cm^(2)` each
Find the value of X.

Options

A. 10
B. 12
C. 8
D. 6

Explanation

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the problem
We have an apple pie with a radius of \( r \) cm. Initially, it is cut into \( x \) identical pieces, each having an area of \( 0.77 \, \text{cm}^2 \). However, it is later discovered that 50% of the pie is rotten, so the remaining 50% is cut into \( x - 3 \) pieces, each with an area of \( 0.616 \, \text{cm}^2 \).

### Step 2: Write the equations
1. The area of the entire pie is given by:
\[ \text{Area} = \pi r^2 \]

2. When the pie is cut into \( x \) pieces, the area of each piece is:
\[ \frac{\pi r^2}{x} = 0.77 \quad \text{(Equation 1)} \]

3. Since 50% of the pie is rotten, the area of the remaining pie is:
\[ \frac{\pi r^2}{2} \]
This remaining area is cut into \( x - 3 \) pieces, so the area of each piece is:
\[ \frac{\frac{\pi r^2}{2}}{x - 3} = 0.616 \quad \text{(Equation 2)} \]

### Step 3: Solve for \( \pi r^2 \)
From Equation 1:
\[ \pi r^2 = 0.77x \]

From Equation 2:
\[ \frac{\pi r^2}{2} = 0.616(x - 3) \]
Multiplying both sides by 2 gives:
\[ \pi r^2 = 1.232(x - 3) \]

### Step 4: Set the equations equal to each other
Now we have two expressions for \( \pi r^2 \):
\[ 0.77x = 1.232(x - 3) \]

### Step 5: Expand and simplify
Expanding the right side:
\[ 0.77x = 1.232x - 3.696 \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 1.232x - 0.77x = 3.696 \]
\[ 0.462x = 3.696 \]

### Step 7: Solve for \( x \)
Dividing both sides by 0.462:
\[ x = \frac{3.696}{0.462} \approx 8 \]

### Conclusion
The value of \( x \) is \( 8 \).