Area of `1^(st)` circle and circumference of IInd circle is 1386 cm2 and 176 cm respectively. There is a square whose side is `35(5)/(7)%` of twice of 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 need to find the perimeter of a square based on the given information about the circles. Let's break it down:

### Step 1: Find the radius of the first circle
The area of the first circle is given as \( 1386 \, \text{cm}^2 \). The formula for the area of a circle is:
\[ A = \pi r^2 \]
where \( r \) is the radius. We can rearrange this formula to find the radius:
\[ r^2 = \frac{A}{\pi} \]
Substituting the area:
\[ r^2 = \frac{1386}{\pi} \]
Using \( \pi \approx 3.14 \):
\[ r^2 \approx \frac{1386}{3.14} \approx 442.04 \]
Taking the square root:
\[ r_1 \approx \sqrt{442.04} \approx 21 \, \text{cm} \]

### Step 2: Find the radius of the second circle
The circumference of the second circle is given as \( 176 \, \text{cm} \). The formula for the circumference of a circle is:
\[ C = 2\pi r \]
We can rearrange this formula to find the radius:
\[ r = \frac{C}{2\pi} \]
Substituting the circumference:
\[ r_2 = \frac{176}{2\pi} = \frac{176}{2 \times 3.14} \approx \frac{176}{6.28} \approx 28 \, \text{cm} \]

### Step 3: Calculate the sum of the radii
Now we can find the sum of the radii of both circles:
\[ \text{Sum of radii} = r_1 + r_2 = 21 + 28 = 49 \, \text{cm} \]

### Step 4: Calculate twice the sum of the radii
Next, we calculate twice the sum of the radii:
\[ 2 \times \text{Sum of radii} = 2 \times 49 = 98 \, \text{cm} \]

### 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 decimal:
\[ 35\frac{5}{7} = 35 + \frac{5}{7} = \frac{245 + 5}{7} = \frac{250}{7} \approx 35.71\% \]
Now, convert this percentage to a decimal:
\[ \frac{35.71}{100} = 0.3571 \]
Now calculate the side of the square:
\[ \text{Side} = 0.3571 \times 98 \approx 35 \, \text{cm} \]

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

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

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