If the circumference of circle is 88 cm and ratio of radius of circle to side of square is 1 : 2 then what will be the ratio of area of circle to area of square.

Options

A. `14:11 `
B. `11:14 `
C. `13:14`
D. `11:16 `

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the given information
We are given:
- Circumference of the circle = 88 cm
- Ratio of the radius of the circle to the side of the square = 1:2

### Step 2: Use the formula for the circumference of a circle
The formula for the circumference \(C\) of a circle is:
\[ C = 2\pi r \]
where \(r\) is the radius of the circle. We can set up the equation:
\[ 2\pi r = 88 \]

### Step 3: Solve for the radius \(r\)
Substituting \(\pi\) with \(\frac{22}{7}\):
\[ 2 \times \frac{22}{7} \times r = 88 \]
Simplifying this:
\[ \frac{44}{7} r = 88 \]
Multiplying both sides by \(\frac{7}{44}\):
\[ r = 88 \times \frac{7}{44} \]
Calculating:
\[ r = 88 \times \frac{7}{44} = 14 \text{ cm} \]

### Step 4: Find the side of the square \(a\)
According to the ratio of the radius to the side of the square:
\[ \frac{r}{a} = \frac{1}{2} \]
Substituting \(r = 14\):
\[ \frac{14}{a} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 14 \times 2 = 1 \times a \implies a = 28 \text{ cm} \]

### Step 5: Calculate the area of the circle
The area \(A_c\) of the circle is given by:
\[ A_c = \pi r^2 \]
Substituting the values:
\[ A_c = \frac{22}{7} \times (14)^2 = \frac{22}{7} \times 196 = \frac{4312}{7} \text{ cm}^2 \]

### Step 6: Calculate the area of the square
The area \(A_s\) of the square is given by:
\[ A_s = a^2 \]
Substituting the value of \(a\):
\[ A_s = (28)^2 = 784 \text{ cm}^2 \]

### Step 7: Find the ratio of the area of the circle to the area of the square
The ratio \(R\) of the area of the circle to the area of the square is:
\[ R = \frac{A_c}{A_s} = \frac{\frac{4312}{7}}{784} \]
Simplifying this:
\[ R = \frac{4312}{7 \times 784} = \frac{4312}{5488} \]
Dividing both the numerator and the denominator by 44:
\[ R = \frac{98}{124} = \frac{49}{62} \]

### Final Answer
The ratio of the area of the circle to the area of the square is:
\[ \frac{49}{62} \]