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�ã−cƒÎ, 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?
Join the centers of the 3 triangles to form an equilateral triangle with sides length 20.
The area enclosed by the 3 triangles will be the area of this triangle minus the area
of 3 sectors of the circles with central angles of 60 degrees. Thus the area of the
enclosed region will be
(1/2)*(20)*(20*sin(60)) - 3*(60/360)*pi*10^2 =
200*(sqrt(3) /2) - 50*pi = 100*sqrt(3) - 50*pi.
So a + b + c = 100 + 3 + 50 = 153.
To find the area enclosed by the three circles, we can break down the problem into smaller parts.
First, let's consider two of the circles that are tangent to each other. The distance between their centers would be equal to the sum of their radii, which is 10 + 10 = 20.
Draw a line connecting the centers of these two circles. This line forms an equilateral triangle with the segment connecting the two points of tangency. Since the circles are equal and tangent to each other, the sides of the equilateral triangle would also measure 20.
Now, let's find the area of this equilateral triangle. The formula for the area of an equilateral triangle with side length s is A = (sqrt(3) / 4) * s^2. In this case, s = 20, so the area of the equilateral triangle formed by two circles is A = (sqrt(3) / 4) * (20^2) = 100 * sqrt(3).
Next, let's consider the third circle. It is tangent to the other two circles and the sides of the equilateral triangle. Drawing lines connecting the centers of the circles, we can see that the center of the third circle forms an equilateral triangle with the centers of the other two circles. This equilateral triangle has side length 20.
To find the area of this equilateral triangle, we use the same formula as before: A = (sqrt(3) / 4) * s^2. In this case, s = 20, so the area of the equilateral triangle formed by the third circle is A = (sqrt(3) / 4) * (20^2) = 100 * sqrt(3).
Finally, to find the area enclosed by the three circles, we subtract the area of the triangle formed by two circles from the area of the triangle formed by the third circle. The area enclosed by the three circles is (100 * sqrt(3)) - (100 * sqrt(3)) = 0.
Therefore, a = 0, b = 0, and c = 0. The value of a + b + c is 0.