Two overlapping identical circles have radii of 6.17. The area of their overlapping sections is 42.7. How far apart are the centers of the circles?

Answer is 6.55

Draw a diagram.

If the intersection points subtend an angle of θ, then the area of overlap is

2*(1/2)r^2(θ-sinθ)
so, you have

6.17^2 (θ-sinθ) = 42.7
θ = 2

The distance between centers is then

2rcos(θ/2) = 2*6.17*cos(1) = 6.67

To find the distance between the centers of two overlapping circles, you can use the Pythagorean theorem. Let's consider the centers of the circles as Point A and Point B, and the distance between them as 'd'.

Since the circles are identical and have radii of 6.17, the distance between each circle's center and the point where they overlap is also 6.17.

Now, draw a line connecting the centers of both circles, forming a right triangle with sides '6.17' and 'd/2'. The length of the hypotenuse of this triangle is 'd', which represents the distance between the centers of the circles.

To find 'd', we can use the Pythagorean theorem:

d^2 = (d/2)^2 + 6.17^2

Simplifying the equation:

d^2 = (d^2)/4 + 6.17^2

Multiply both sides of the equation by 4 to eliminate the fraction:

4d^2 = d^2 + 4 * 6.17^2

Applying the distributive property:

4d^2 = d^2 + 4 * 38.0569

4d^2 = d^2 + 152.2276

Subtract d^2 from both sides of the equation:

3d^2 = 152.2276

Divide both sides of the equation by 3:

d^2 = 50.74253333

Taking the square root of both sides:

d ≈ √(50.74253333)

d ≈ 7.12

Since the distance calculated is the hypotenuse, which is twice the distance between the center and the point of overlap, we divide by 2:

Distance between the centers = d/2 ≈ 7.12/2 ≈ 3.56

Therefore, the distance between the centers of the overlapping circles is approximately 3.56. However, the given answer is 6.55, which suggests that there might be a mistake in the calculations or information provided.