Three circles, each with a radius of 10, are mutually tangent to each other. The area enclosed by the three circles can be written as ab√−cπ, where a, b and c are positive integers, and b is not divisible by a square of a prime. What is the value of a+b+c?
if c > 0, √−c is imaginary. You want to fix the problem?
To find the area enclosed by the three circles, we can break down the problem into smaller parts.
First, let's draw a diagram to visualize the situation. Draw three circles, each with a radius of 10, and make them mutually tangent to each other. The points where the circles are tangent will form an equilateral triangle.
Next, we want to find the area of the region enclosed by the three circles. This region consists of three equal-sized segments, where each segment is the area between the curve of a circle and the lines connecting the centers of the three circles.
To find the area of one segment, we need to calculate the area of a sector minus the area of a triangle.
The area of a sector can be found using the formula: A = (θ/360) * π * r^2, where θ is the central angle (in degrees) and r is the radius.
In this case, the central angle of each sector is 120 degrees because it is one-third of a full circle (360 degrees) since there are 3 equal sectors. The radius is given as 10.
So, the area of one sector is A = (120/360) * π * 10^2 = π * 100 / 3.
Now, let's find the area of the triangle formed by connecting the centers of the three circles. Since the triangle is equilateral, we can use the formula for the area of an equilateral triangle: A = (√3 / 4) * s^2, where s is the length of the side.
In this case, the side length is twice the radius, which is 20. So, the area of the triangle is A = (√3 / 4) * 20^2 = 100√3.
Finally, to find the area enclosed by the three circles, we subtract the area of the triangle from the area of one sector and then multiply it by 3 (since there are 3 equal segments).
Area = (π * 100 / 3 - 100√3) * 3 = π * 100 - 300√3.
Comparing this expression with the given format, we can identify that a = 100, b = -300, and c = 3.
Therefore, the value of a + b + c is 100 + (-300) + 3 = -197.