Suppose we have two circles of radius 2 each in the plane such that the distance between their centres is `2sqrt3.` The area of the region common to both circles lies between

Options

A. 0.5 and 0.6
B. 0.65 and 0.7
C. 0.7 and 0.75
D. 0.8 and 0.9

Explanation

To find the area of the region common to both circles, we can follow these steps:

### Step 1: Understand the configuration of the circles
We have two circles, each with a radius \( r = 2 \). The distance between their centers is given as \( d = 2\sqrt{3} \).

### Step 2: Check the relationship between the distance and the radii
To determine if the circles overlap, we need to check if the distance \( d \) is less than the sum of the radii. The sum of the radii is:
\[ r + r = 2 + 2 = 4 \]
Since \( d = 2\sqrt{3} \approx 3.464 \), which is less than 4, the circles do overlap.

### Step 3: Use the formula for the area of intersection of two circles
The area \( A \) of the intersection of two circles can be calculated using the formula:
\[ A = 2r^2 \cos^{-1} \left( \frac{d}{2r} \right) - \frac{d}{2} \sqrt{4r^2 - d^2} \]
where \( r \) is the radius of the circles and \( d \) is the distance between their centers.

### Step 4: Substitute the values into the formula
Here, \( r = 2 \) and \( d = 2\sqrt{3} \):
\[ A = 2(2^2) \cos^{-1} \left( \frac{2\sqrt{3}}{2 \cdot 2} \right) - \frac{2\sqrt{3}}{2} \sqrt{4(2^2) - (2\sqrt{3})^2} \]
This simplifies to:
\[ A = 8 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) - \sqrt{3} \sqrt{16 - 12} \]

### Step 5: Calculate \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \)
We know that:
\[ \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \]
So we substitute this into the area formula:
\[ A = 8 \cdot \frac{\pi}{6} - \sqrt{3} \cdot 2 \]
This simplifies to:
\[ A = \frac{4\pi}{3} - 2\sqrt{3} \]

### Step 6: Calculate the numerical values
Now we can approximate the values:
- \( \frac{4\pi}{3} \approx 4.18879 \)
- \( 2\sqrt{3} \approx 3.464 \)

Thus:
\[ A \approx 4.18879 - 3.464 \approx 0.72479 \]

### Step 7: Determine the range of the area
Since we have calculated the area of intersection, we can conclude that the area of the region common to both circles lies between:
\[ 0 < A < 4 \]

### Final Answer
The area of the region common to both circles lies between \( 0 \) and \( 4 \).