The area of one circle and the circumference of another circle are 1386 `cm^2` and 176 cm respectively. There is a square whose side is `35,5/7%` of twice the sum of the radius of both the circles. Find the perimeter of the square (in cm).

Options

A. 132
B. 136
C. 140
D. 116

Answer

To solve the problem step by step, we will first find the radii of the two circles using the given area and circumference, then calculate the side of the square based on the radius, and finally find the perimeter of the square.

### Step 1: Find the radius of the first circle using the area.
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Given that the area is \( 1386 \, \text{cm}^2 \):
\[ \pi r^2 = 1386 \]
To find \( r^2 \):
\[ r^2 = \frac{1386}{\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 = \frac{1386}{3.14} \approx 442.04 \]
Now, taking the square root to find \( r \):
\[ r \approx \sqrt{442.04} \approx 21 \, \text{cm} \]

### Step 2: Find the radius of the second circle using the circumference.
The circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
Given that the circumference is \( 176 \, \text{cm} \):
\[ 2\pi r = 176 \]
To find \( r \):
\[ r = \frac{176}{2\pi} = \frac{176}{2 \times 3.14} \approx \frac{176}{6.28} \approx 28 \, \text{cm} \]

### Step 3: Calculate the sum of the radii.
Now we add the radii of both circles:
\[ \text{Sum of radii} = 21 \, \text{cm} + 28 \, \text{cm} = 49 \, \text{cm} \]

### Step 4: Calculate twice the sum of the radii.
\[ \text{Twice the sum of radii} = 2 \times 49 = 98 \, \text{cm} \]

### Step 5: Calculate the side of the square.
The side of the square is given as \( 35 \frac{5}{7}\% \) of twice the sum of the radii. First, convert \( 35 \frac{5}{7}\% \) to a fraction:
\[ 35 \frac{5}{7} = \frac{245 + 5}{7} = \frac{250}{7} \]
Now calculate \( \frac{250}{7}\% \) of \( 98 \):
\[ \text{Side of square} = \frac{250}{7} \times \frac{98}{100} = \frac{250 \times 98}{700} = \frac{24500}{700} = 35 \, \text{cm} \]

### Step 6: Calculate the perimeter of the square.
The perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Thus,
\[ P = 4 \times 35 = 140 \, \text{cm} \]

### Final Answer:
The perimeter of the square is \( 140 \, \text{cm} \).

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