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
If entire pie would have been cut into 11 identical pieces what would have been top surface area of each piece?

Options

A. `0.64 cm^(2)`
B. `0.56 cm^(2)`
C. `0.28 cm^(2)`
D. `0.42 cm^(2)`

Explanation

To solve the problem step by step, we will follow the information provided in the question and derive the necessary equations to find the required area of each piece if the entire pie is cut into 11 identical pieces.

### Step 1: Understand the Area of the Pie
The area of the entire apple pie can be calculated using the formula for the area of a circle:
\[ \text{Area of the pie} = \pi r^2 \]

### Step 2: Set Up the First Equation
Initially, the pie is cut into \(x\) identical pieces, and the area of each piece is given as \(0.77 \, \text{cm}^2\). Therefore, we can write:
\[ \frac{\pi r^2}{x} = 0.77 \quad \text{(Equation 1)} \]

### Step 3: Understand the Effect of Rotting
Since 50% of the pie is rotten, only 50% of the pie is usable. The area of the remaining pie is:
\[ \text{Remaining area} = \frac{\pi r^2}{2} \]

### Step 4: Set Up the Second Equation
The remaining usable area is cut into \(x - 3\) pieces, and the area of each of these pieces is given as \(0.616 \, \text{cm}^2\). Thus, we can write:
\[ \frac{\frac{\pi r^2}{2}}{x - 3} = 0.616 \quad \text{(Equation 2)} \]

### Step 5: Simplify Equation 2
Multiplying both sides of Equation 2 by \(x - 3\) gives:
\[ \frac{\pi r^2}{2} = 0.616(x - 3) \]

### Step 6: Relate the Two Equations
Now we have two equations:
1. \(\frac{\pi r^2}{x} = 0.77\)
2. \(\frac{\pi r^2}{2} = 0.616(x - 3)\)

From Equation 1, we can express \(\pi r^2\):
\[ \pi r^2 = 0.77x \]

Substituting this into Equation 2:
\[ \frac{0.77x}{2} = 0.616(x - 3) \]

### Step 7: Solve for \(x\)
Multiplying through by 2 to eliminate the fraction:
\[ 0.77x = 1.232(x - 3) \]
Expanding the right side:
\[ 0.77x = 1.232x - 3.696 \]
Rearranging gives:
\[ 1.232x - 0.77x = 3.696 \]
\[ 0.462x = 3.696 \]
\[ x = \frac{3.696}{0.462} \approx 8 \]

### Step 8: Calculate the Total Area of the Pie
Now that we have \(x = 8\), we can substitute back into Equation 1 to find the total area of the pie:
\[ \pi r^2 = 0.77 \times 8 = 6.16 \, \text{cm}^2 \]

### Step 9: Find the Area of Each Piece if Cut into 11 Pieces
If the entire pie is cut into 11 identical pieces, the area of each piece would be:
\[ \text{Area of each piece} = \frac{\pi r^2}{11} = \frac{6.16}{11} \approx 0.56 \, \text{cm}^2 \]

### Final Answer
The top surface area of each piece if the entire pie is cut into 11 identical pieces is approximately \(0.56 \, \text{cm}^2\).