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 a√b−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?
Draw lines joining the centers of the circles. That is an equilateral triangle with side 10.
Area of triangle is 1/2 * 5 * 5√3 = 25/2 √3
Now, the angles of the triangle are all π/3, so, since the area of a sector of a circle is a = 1/2 r^2 θ, each sector has area 1/2 * 25 * π/3 = 25/6 π
So, the area in between all the circles is the area of the triangle, less the total area of the 3 sectors.
I think you can take it from here.
Oops. I was thinking diameter of 10, not radius. Adjust calculations accordingly.
To find the area enclosed by the three circles, we can start by visualizing the setup of the circles.
Since the three circles are mutually tangent to each other, they form an equilateral triangle with sides equal to the sum of the diameters of the circles (which is 20).
We can divide the triangle into three smaller triangles, each of which has a central angle of 60 degrees. Using trigonometry, we can calculate the height of each smaller triangle. Let's call the height 'h'.
Using the properties of an equilateral triangle, we know that the height of an equilateral triangle is given by the formula:
h = (sqrt(3)/2) * side
In this case, the side of the equilateral triangle is 20, so:
h = (sqrt(3)/2) * 20
= 10sqrt(3)
Since each smaller triangle is an isosceles triangle, with a base equal to the diameter of a circle (which is 20), we can calculate the area of each smaller triangle using the formula:
Area of triangle = (1/2) * base * height
Area of triangle = (1/2) * 20 * 10sqrt(3)
= 100sqrt(3)
Now, since there are three smaller triangles, the total area enclosed by the three circles is:
Total area = 3 * Area of triangle
= 3 * 100sqrt(3)
= 300sqrt(3)
Therefore, a = 300, b = 3, and c = 0.
Thus, the value of a+b+c is 300 + 3 + 0 = 303.