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 circumference of the original pie.

Options

A. `(44)/(5) cm`
B. `(88)/(25) cm`
C. `(176)/(25) cm`
D. `(176)/(50) cm `

Explanation

To solve the problem step by step, we will use the information provided and apply some mathematical reasoning.

### Step 1: Calculate the total area of the original apple pie.
The area \( A \) of a circle (the top surface of the apple pie) is given by the formula:
\[ A = \pi r^2 \]

### Step 2: Set up the equation for the original pie.
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 express the total area of the pie as:
\[ \frac{\pi r^2}{x} = 0.77 \quad \text{(Equation 1)} \]

### Step 3: Calculate the area of the remaining pie after 50% is rotten.
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, with each piece having an area of \( 0.616 \, \text{cm}^2 \). Thus, we can set up the second equation:
\[ \frac{\pi r^2 / 2}{x - 3} = 0.616 \quad \text{(Equation 2)} \]

### Step 4: Rearranging Equation 1 and Equation 2.
From Equation 1:
\[ \pi r^2 = 0.77x \quad \text{(1')} \]
From Equation 2:
\[ \pi r^2 = 1.232(x - 3) \quad \text{(2')} \]

### Step 5: Set the equations equal to each other.
Since both expressions equal \( \pi r^2 \), we can set them equal:
\[ 0.77x = 1.232(x - 3) \]

### Step 6: Solve for \( x \).
Expanding the right side:
\[ 0.77x = 1.232x - 3.696 \]
Rearranging gives:
\[ 1.232x - 0.77x = 3.696 \]
\[ 0.462x = 3.696 \]
\[ x = \frac{3.696}{0.462} = 8 \]

### Step 7: Substitute \( x \) back into Equation 1 to find \( r \).
Substituting \( x = 8 \) into Equation 1:
\[ \pi r^2 = 0.77 \times 8 \]
\[ \pi r^2 = 6.16 \]
\[ r^2 = \frac{6.16}{\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{6.16 \times 7}{22} = \frac{43.12}{22} \approx 1.96 \]
Taking the square root:
\[ r \approx \sqrt{1.96} \approx 1.4 \, \text{cm} \]

### Step 8: Calculate the circumference of the original pie.
The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Substituting \( r = 1.4 \):
\[ C = 2 \times \frac{22}{7} \times 1.4 \]
Calculating gives:
\[ C = \frac{44}{7} \approx 8.8 \, \text{cm} \]

### Final Answer:
The circumference of the original pie is \( \frac{44}{5} \, \text{cm} \) or \( 8.8 \, \text{cm} \).

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