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?

153

153

To find the area enclosed by the three circles, we can draw a triangle connecting their centers.

Since the circles are mutually tangent, the triangle formed will be an equilateral triangle.

To find the area of the equilateral triangle, we need to calculate its side length.

In an equilateral triangle, all sides are equal. Therefore, we can use the radius of one circle as the side length of the triangle.

The formula to calculate the area of an equilateral triangle is:

Area = (sqrt(3)/4) * side length^2

Substituting the value of the side length as the radius of one circle (10), we get:

Area = (sqrt(3)/4) * 10^2

Area = (sqrt(3)/4) * 100

Simplifying the expression, we have:

Area = 25sqrt(3)

Since the area enclosed by the three circles is equal to the area of the equilateral triangle, the area can be written as:

ab√−cπ = 25sqrt(3)

Comparing the equation with the given format, we can deduce that:

a = 25, b = sqrt(3), and c = 0

As b is not divisible by a square of a prime, b = sqrt(3).

Therefore, a + b + c = 25 + sqrt(3) + 0 = 25 + sqrt(3).

Finally, the value of a + b + c is 25 + sqrt(3).